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Divisor function

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</style><div role="note" class="hatnote navigation-not-searchable">"Robin's theorem" redirects here. For Robbins' theorem in graph theory, see <a href="Robbins'_theorem" title="Robbins' theorem">Robbins' theorem</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, and specifically in <a href="Number_theory" title="Number theory">number theory</a>, a divisor function is an <a href="Arithmetic_function" title="Arithmetic function">arithmetic function</a> related to the <a href="Divisor" title="Divisor">divisors</a> of an <a href="Integer" title="Integer">integer</a>. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a> and the <a href="Eisenstein_series" title="Eisenstein series">Eisenstein series</a> of <a href="Modular_form" title="Modular form">modular forms</a>. Divisor functions were studied by <a href="Ramanujan" class="mw-redirect" title="Ramanujan">Ramanujan</a>, who gave a number of important <a href="Modular_arithmetic" title="Modular arithmetic">congruences</a> and <a href="Identity_(mathematics)" title="Identity (mathematics)">identities</a>; these are treated separately in the article <a href="Ramanujan's_sum" title="Ramanujan's sum">Ramanujan's sum</a>.

A related function is the <a href="Divisor_summatory_function" title="Divisor summatory function">divisor summatory function</a>, which, as the name implies, is a sum over the divisor function.

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

The sum of positive divisors function σz(n), for a real or complex number z, is defined as the <a href="Summation" title="Summation">sum</a> of the zth <a href="Exponentiation" title="Exponentiation">powers</a> of the positive <a href="Divisor" title="Divisor">divisors</a> of n. It can be expressed in <a href="Summation#Capital-sigma_notation" title="Summation">sigma notation</a> as

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{z}(n)=\sum _{d\mid n}d^{z}\,\!,}</annotation>

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</math><img src="./31f7ac1f3734c4f943b6c22f44a81cdd1b91bda0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:15.24ex; height:6.009ex;" alt="{\displaystyle \sigma _{z}(n)=\sum _{d\mid n}d^{z}\,\!,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {d\mid n}}">

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<annotation encoding="application/x-tex">{\displaystyle {d\mid n}}</annotation>

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</math><img src="./2185b50b396d82f566e1b747e3f1ffbe5c92fcd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.548ex; height:2.843ex;" alt="{\displaystyle {d\mid n}}" loading="lazy"> is shorthand for "d <a href="Divides" class="mw-redirect" title="Divides">divides</a> n".

The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ0(n), or the number-of-divisors function<a href="#cite_note-Long_1972_46-1">[1]</a><a href="#cite_note-2">[2]</a> (<a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A000005" class="extiw external" title="oeis:A000005">A000005</a>). When z is 1, the function is called the sigma function or sum-of-divisors function,<a href="#cite_note-Long_1972_46-1">[1]</a><a href="#cite_note-3">[3]</a> and the subscript is often omitted, so σ(n) is the same as σ1(n) (<a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A000203" class="extiw external" title="oeis:A000203">A000203</a>).

The <a href="Aliquot_sum" title="Aliquot sum">aliquot sum</a> s(n) of n is the sum of the <a href="Proper_divisor" class="mw-redirect" title="Proper divisor">proper divisors</a> (that is, the divisors excluding n itself, <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A001065" class="extiw external" title="oeis:A001065">A001065</a>), and equals σ1(n) − n; the <a href="Aliquot_sequence" title="Aliquot sequence">aliquot sequence</a> of n is formed by repeatedly applying the aliquot sum function.

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

For example, σ0(12) is the number of the divisors of 12:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{0}(12)&=1^{0}+2^{0}+3^{0}+4^{0}+6^{0}+12^{0}\\&=1+1+1+1+1+1=6,\end{aligned}}}</annotation>

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</math><img src="./549ef378f213e9de114b5edf119f4d236dd6f965.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.03ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{0}(12)&=1^{0}+2^{0}+3^{0}+4^{0}+6^{0}+12^{0}\\&=1+1+1+1+1+1=6,\end{aligned}}}" loading="lazy"></dd></dl>

while σ1(12) is the sum of all the divisors:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{1}(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}+12^{1}\\&=1+2+3+4+6+12=28,\end{aligned}}}</annotation>

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</math><img src="./e84ff27cd7cd28822f3e3ee0235f86d1859ff0b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.03ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{1}(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}+12^{1}\\&=1+2+3+4+6+12=28,\end{aligned}}}" loading="lazy"></dd></dl>

and the aliquot sum s(12) of proper divisors is:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}s(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}\\&=1+2+3+4+6=16.\end{aligned}}}</annotation>

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</math><img src="./3aeb100fd2676c461a1312b5c606f90aaeb65345.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:32.319ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}s(12)&=1^{1}+2^{1}+3^{1}+4^{1}+6^{1}\\&=1+2+3+4+6=16.\end{aligned}}}" loading="lazy"></dd></dl>

σ−1(n) is sometimes called the <a href="Abundancy_index" class="mw-redirect" title="Abundancy index">abundancy index</a> of n, and we have:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{-1}(12)&=1^{-1}+2^{-1}+3^{-1}+4^{-1}+6^{-1}+12^{-1}\\[6pt]&={\tfrac {1}{1}}+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{12}}\\[6pt]&={\tfrac {12}{12}}+{\tfrac {6}{12}}+{\tfrac {4}{12}}+{\tfrac {3}{12}}+{\tfrac {2}{12}}+{\tfrac {1}{12}}\\[6pt]&={\tfrac {12+6+4+3+2+1}{12}}={\tfrac {28}{12}}={\tfrac {7}{3}}={\tfrac {\sigma _{1}(12)}{12}}\end{aligned}}}</annotation>

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</math><img src="./88918bd7732280a37e9aa6924d06cc1b901dd3c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.848ex; margin-bottom: -0.323ex; width:47.98ex; height:19.509ex;" alt="{\displaystyle {\begin{aligned}\sigma _{-1}(12)&=1^{-1}+2^{-1}+3^{-1}+4^{-1}+6^{-1}+12^{-1}\\[6pt]&={\tfrac {1}{1}}+{\tfrac {1}{2}}+{\tfrac {1}{3}}+{\tfrac {1}{4}}+{\tfrac {1}{6}}+{\tfrac {1}{12}}\\[6pt]&={\tfrac {12}{12}}+{\tfrac {6}{12}}+{\tfrac {4}{12}}+{\tfrac {3}{12}}+{\tfrac {2}{12}}+{\tfrac {1}{12}}\\[6pt]&={\tfrac {12+6+4+3+2+1}{12}}={\tfrac {28}{12}}={\tfrac {7}{3}}={\tfrac {\sigma _{1}(12)}{12}}\end{aligned}}}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Table_of_values">Table of values</h2></div>

The cases x = 2 to 5 are listed in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A001157" class="extiw external" title="oeis:A001157">A001157</a> through <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A001160" class="extiw external" title="oeis:A001160">A001160</a>, x = 6 to 24 are listed in <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A013954" class="extiw external" title="oeis:A013954">A013954</a> through <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A013972" class="extiw external" title="oeis:A013972">A013972</a>.

<th>prime factorization</th>

<th>𝜎4(n)

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<td>1

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<td>17

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<td>82

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<td>273

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<td>626

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<td>1394

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<td>2402

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<td>4369

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<td>6643

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<td>10642

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<td>14642

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<td>22386

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<td>28562

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<td>40834

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<td>51332

</td></tr>

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<td>69905

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<td>83522

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<tr>

<td>112931

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<td>130322

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<tr>

<td>170898

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<td>196964

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<td>248914

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<td>279842

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<td>358258

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<td>391251

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<td>485554

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<td>538084

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<td>655746

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<td>707282

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<td>872644

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<td>923522

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<td>1118481

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<td>1200644

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<td>1419874

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<td>1503652

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<td>1813539

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<td>1874162

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<td>2215474

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<td>2342084

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<td>2734994

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<td>2825762

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<td>3348388

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<td>3418802

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<td>3997266

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<td>4158518

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<td>4757314

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<td>4879682

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<td>5732210

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<td>5767203

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<td>6651267

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<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Formulas_at_prime_powers">Formulas at prime powers</h3></div>

For a <a href="Prime_number" title="Prime number">prime number</a> p,

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{0}(p)&=2\\\sigma _{0}(p^{n})&=n+1\\\sigma _{1}(p)&=p+1\end{aligned}}}</annotation>

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</math><img src="./fde3566adee8361928b15c0f4d5442a51ebf49b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:15.826ex; height:9.176ex;" alt="{\displaystyle {\begin{aligned}\sigma _{0}(p)&=2\\\sigma _{0}(p^{n})&=n+1\\\sigma _{1}(p)&=p+1\end{aligned}}}" loading="lazy"></dd></dl>

because by definition, the factors of a prime number are 1 and itself. Also, where pn# denotes the <a href="Primorial" title="Primorial">primorial</a>,

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}(p_{n}\#)=2^{n}}</annotation>

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</math><img src="./346590c3523116ddcdd0e9c611a76e32fcd25bc3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.994ex; height:2.843ex;" alt="{\displaystyle \sigma _{0}(p_{n}\#)=2^{n}}" loading="lazy"></dd></dl>

since n prime factors allow a sequence of binary selection (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{i}}">

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</math><img src="./5bab39399bf5424f25d957cdc57c84a0622626d2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.059ex; height:2.009ex;" alt="{\displaystyle p_{i}}" loading="lazy"> or 1) from n terms for each proper divisor formed. However, these are not in general the smallest numbers whose number of divisors is a <a href="Power_of_two" title="Power of two">power of two</a>; instead, the smallest such number may be obtained by multiplying together the first n <a href="Fermi%E2%80%93Dirac_prime" title="Fermi–Dirac prime">Fermi–Dirac primes</a>, prime powers whose exponent is a power of two.<a href="#cite_note-4">[4]</a>

Clearly, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1<\sigma _{0}(n)<n}">

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<annotation encoding="application/x-tex">{\displaystyle 1<\sigma _{0}(n)<n}</annotation>

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</math><img src="./52a0cf797911b12eca8963901c61995e9b517dad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.34ex; height:2.843ex;" alt="{\displaystyle 1<\sigma _{0}(n)<n}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>2}">

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<annotation encoding="application/x-tex">{\displaystyle n>2}</annotation>

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</math><img src="./44e71ac55b9fbf1e9f341b946cda63d61d3ef2cd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>2}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{x}(n)>n}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{x}(n)>n}</annotation>

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</math><img src="./3d7dd9bc839a48b52099db270fee212003c69619.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.197ex; height:2.843ex;" alt="{\displaystyle \sigma _{x}(n)>n}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>1}">

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<annotation encoding="application/x-tex">{\displaystyle n>1}</annotation>

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</math><img src="./ee74e1cc07e7041edf0fcbd4481f5cd32ad17b64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>1}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x>0}">

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<annotation encoding="application/x-tex">{\displaystyle x>0}</annotation>

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</math><img src="./80d24be5f0eb4a9173da6038badc8659546021d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x>0}" loading="lazy"> .

The divisor function is <a href="Multiplicative_function" title="Multiplicative function">multiplicative</a> (since each divisor c of the product mn with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gcd(m,n)=1}">

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<annotation encoding="application/x-tex">{\displaystyle \gcd(m,n)=1}</annotation>

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</math><img src="./f43c4052a27ae23281e912d3e0b50357ff5f16a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.027ex; height:2.843ex;" alt="{\displaystyle \gcd(m,n)=1}" loading="lazy"> distinctively correspond to a divisor a of m and a divisor b of n), but not <a href="Completely_multiplicative_function" title="Completely multiplicative function">completely multiplicative</a>:

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<annotation encoding="application/x-tex">{\displaystyle \gcd(a,b)=1\Longrightarrow \sigma _{x}(ab)=\sigma _{x}(a)\sigma _{x}(b).}</annotation>

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</math><img src="./dc5b915c943572ca83de930a3b0ccdfe42c89dd7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.042ex; height:2.843ex;" alt="{\displaystyle \gcd(a,b)=1\Longrightarrow \sigma _{x}(ab)=\sigma _{x}(a)\sigma _{x}(b).}" loading="lazy"></dd></dl>

The consequence of this is that, if we write

</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=\prod _{i=1}^{r}p_{i}^{a_{i}}}</annotation>

</semantics>

</math><img src="./838611821c9b7161f6f5db4db73a4058a51d757b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:10.746ex; height:6.843ex;" alt="{\displaystyle n=\prod _{i=1}^{r}p_{i}^{a_{i}}}" loading="lazy"></dd></dl>

where r = ω(n) is the <a href="Prime_omega_function" title="Prime omega function">number of distinct prime factors</a> of n, pi is the ith prime factor, and ai is the maximum power of pi by which n is <a href="Divisible" class="mw-redirect" title="Divisible">divisible</a>, then we have: <a href="#cite_note-FOOTNOTEHardyWright2008310_f§16.7-5">[5]</a>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</munderover>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</munderover>

<mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{x}(n)=\prod _{i=1}^{r}\sum _{j=0}^{a_{i}}p_{i}^{jx}=\prod _{i=1}^{r}\left(1+p_{i}^{x}+p_{i}^{2x}+\cdots +p_{i}^{a_{i}x}\right).}</annotation>

</semantics>

</math><img src="./5edb3de0c1bd4b8013904d328941dd7f55e7aaa4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:53.129ex; height:7.343ex;" alt="{\displaystyle \sigma _{x}(n)=\prod _{i=1}^{r}\sum _{j=0}^{a_{i}}p_{i}^{jx}=\prod _{i=1}^{r}\left(1+p_{i}^{x}+p_{i}^{2x}+\cdots +p_{i}^{a_{i}x}\right).}" loading="lazy"></dd></dl>

which, when x ≠ 0, is equivalent to the useful formula: <a href="#cite_note-FOOTNOTEHardyWright2008310_f§16.7-5">[5]</a>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<mfrac>

<mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{x}(n)=\prod _{i=1}^{r}{\frac {p_{i}^{(a_{i}+1)x}-1}{p_{i}^{x}-1}}.}</annotation>

</semantics>

</math><img src="./7b304c810f7c9f6f974d7815e4ca985c1e5d4833.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.861ex; height:7.676ex;" alt="{\displaystyle \sigma _{x}(n)=\prod _{i=1}^{r}{\frac {p_{i}^{(a_{i}+1)x}-1}{p_{i}^{x}-1}}.}" loading="lazy"></dd></dl>

When x = 0, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{0}(n)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}(n)}</annotation>

</semantics>

</math><img src="./db40844785581334712908eaf6f13c9a23eaae8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.586ex; height:2.843ex;" alt="{\displaystyle \sigma _{0}(n)}" loading="lazy"> is: <a href="#cite_note-FOOTNOTEHardyWright2008310_f§16.7-5">[5]</a>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}(n)=\prod _{i=1}^{r}(a_{i}+1).}</annotation>

</semantics>

</math><img src="./1ad40b3f0647389bf3ba70c2cc7987a5448fa3dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.142ex; height:6.843ex;" alt="{\displaystyle \sigma _{0}(n)=\prod _{i=1}^{r}(a_{i}+1).}" loading="lazy"></dd></dl>

This result can be directly deduced from the fact that all divisors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> are uniquely determined by the distinct tuples <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2},...,x_{i},...,x_{r})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2},...,x_{i},...,x_{r})}</annotation>

</semantics>

</math><img src="./12711f3e712373f71ac699ba291f7c884b158a0e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.383ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2},...,x_{i},...,x_{r})}" loading="lazy"> of integers with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq x_{i}\leq a_{i}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq x_{i}\leq a_{i}}</annotation>

</semantics>

</math><img src="./0dbcde5c366b4ac1dd558e31c0db886ad090feaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.518ex; height:2.509ex;" alt="{\displaystyle 0\leq x_{i}\leq a_{i}}" loading="lazy"> (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i}+1}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{i}+1}</annotation>

</semantics>

</math><img src="./77b996b29b4b1cb33239b773668e1123758cdeba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.032ex; height:2.509ex;" alt="{\displaystyle a_{i}+1}" loading="lazy"> independent choices for each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

</semantics>

</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy">).

For example, if n is 24, there are two prime factors (p1 is 2; p2 is 3); noting that 24 is the product of 23×31, a1 is 3 and a2 is 1. Thus we can calculate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{0}(24)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}(24)}</annotation>

</semantics>

</math><img src="./105e935587f5aab066e57c947ce3bbdeef1b33e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.516ex; height:2.843ex;" alt="{\displaystyle \sigma _{0}(24)}" loading="lazy"> as so:

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}(24)=\prod _{i=1}^{2}(a_{i}+1)=(3+1)(1+1)=4\cdot 2=8.}</annotation>

</semantics>

</math><img src="./26e73a8389f8f47e42b75e2227000e9b37ee5f8a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.483ex; height:7.343ex;" alt="{\displaystyle \sigma _{0}(24)=\prod _{i=1}^{2}(a_{i}+1)=(3+1)(1+1)=4\cdot 2=8.}" loading="lazy"></dd></dl>

The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.

<div class="mw-heading mw-heading3"><h3 id="Other_properties_and_identities">Other properties and identities</h3></div>

<a href="Euler" class="mw-redirect" title="Euler">Euler</a> proved the remarkable recurrence:<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a><a href="#cite_note-8">[8]</a>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mrow>

</munder>

<msup>

</mrow>

</msup>

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\sigma _{1}(n)&=\sigma _{1}(n-1)+\sigma _{1}(n-2)-\sigma _{1}(n-5)-\sigma _{1}(n-7)+\sigma _{1}(n-12)+\sigma _{1}(n-15)+\cdots \\[12mu]&=\sum _{i\in \mathbb {N} }(-1)^{i+1}\left(\sigma _{1}\left(n-{\frac {1}{2}}\left(3i^{2}-i\right)\right)+\sigma _{1}\left(n-{\frac {1}{2}}\left(3i^{2}+i\right)\right)\right),\end{aligned}}}</annotation>

</semantics>

</math><img src="./57edf7ce8b5ba7cd703d6696d6272fc4bce20ffc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:89.057ex; height:11.509ex;" alt="{\displaystyle {\begin{aligned}\sigma _{1}(n)&=\sigma _{1}(n-1)+\sigma _{1}(n-2)-\sigma _{1}(n-5)-\sigma _{1}(n-7)+\sigma _{1}(n-12)+\sigma _{1}(n-15)+\cdots \\[12mu]&=\sum _{i\in \mathbb {N} }(-1)^{i+1}\left(\sigma _{1}\left(n-{\frac {1}{2}}\left(3i^{2}-i\right)\right)+\sigma _{1}\left(n-{\frac {1}{2}}\left(3i^{2}+i\right)\right)\right),\end{aligned}}}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}(0)=n}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{1}(0)=n}</annotation>

</semantics>

</math><img src="./54ce45d1d1b4c6123b03d5a161a6aa60786e5016.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.847ex; height:2.843ex;" alt="{\displaystyle \sigma _{1}(0)=n}" loading="lazy"> if it occurs and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}(x)=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{1}(x)=0}</annotation>

</semantics>

</math><img src="./2dfc8296243c6fee4247017dcdb42166b6f0d54d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.781ex; height:2.843ex;" alt="{\displaystyle \sigma _{1}(x)=0}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x<0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x<0}</annotation>

</semantics>

</math><img src="./1a4dbbf970b2d2863dcab589eafe006f08e727d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.591ex; height:2.176ex;" alt="{\displaystyle x<0}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}\left(3i^{2}\mp i\right)}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}\left(3i^{2}\mp i\right)}</annotation>

</semantics>

</math><img src="./7d94fb852b626be4f9c9039c222e48e9223163a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.837ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}\left(3i^{2}\mp i\right)}" loading="lazy"> are consecutive pairs of generalized <a href="Pentagonal_numbers" class="mw-redirect" title="Pentagonal numbers">pentagonal numbers</a> (<a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A001318" class="extiw external" title="oeis:A001318">A001318</a>, starting at offset 1). Indeed, Euler proved this by logarithmic differentiation of the identity in his <a href="Pentagonal_number_theorem" title="Pentagonal number theorem">pentagonal number theorem</a>.

For a non-square integer, n, every divisor, d, of n is paired with divisor n/d of n and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{0}(n)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}(n)}</annotation>

</semantics>

</math><img src="./db40844785581334712908eaf6f13c9a23eaae8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.586ex; height:2.843ex;" alt="{\displaystyle \sigma _{0}(n)}" loading="lazy"> is even; for a square integer, one divisor (namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {n}}}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\sqrt {n}}}</annotation>

</semantics>

</math><img src="./2a2994734eae382ce30100fb17b9447fd8e99f81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.331ex; height:3.009ex;" alt="{\displaystyle {\sqrt {n}}}" loading="lazy">) is not paired with a distinct divisor and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{0}(n)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}(n)}</annotation>

</semantics>

</math><img src="./db40844785581334712908eaf6f13c9a23eaae8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.586ex; height:2.843ex;" alt="{\displaystyle \sigma _{0}(n)}" loading="lazy"> is odd. Similarly, the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma _{1}(n)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma _{1}(n)}</annotation>

</semantics>

</math><img src="./30e6dcdcf20087da1d219fdc818382bcc77f4fbc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.586ex; height:2.843ex;" alt="{\displaystyle \sigma _{1}(n)}" loading="lazy"> is odd if and only if n is a square or twice a square.<a href="#cite_note-FOOTNOTEGioiaVaidya1967-9">[9]</a>

We also note s(n) = σ(n) − n. Here s(n) denotes the sum of the proper divisors of n, that is, the divisors of n excluding n itself. This function is used to recognize <a href="Perfect_number" title="Perfect number">perfect numbers</a>, which are the n such that s(n) = n. If s(n) > n, then n is an <a href="Abundant_number" title="Abundant number">abundant number</a>, and if s(n) < n, then n is a <a href="Deficient_number" title="Deficient number">deficient number</a>.

If n is a power of 2, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=2^{k}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=2^{k}}</annotation>

</semantics>

</math><img src="./a8242689a132230c6c847ae901f6f32dba8181ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.744ex; height:2.676ex;" alt="{\displaystyle n=2^{k}}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (n)=2\cdot 2^{k}-1=2n-1}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (n)=2\cdot 2^{k}-1=2n-1}</annotation>

</semantics>

</math><img src="./f66aac02972358e530ea0620cfe582e9b5076d2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.386ex; height:3.176ex;" alt="{\displaystyle \sigma (n)=2\cdot 2^{k}-1=2n-1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(n)=n-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s(n)=n-1}</annotation>

</semantics>

</math><img src="./3c53dadaca0aff79f0306461c4870ab70e49eb5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.79ex; height:2.843ex;" alt="{\displaystyle s(n)=n-1}" loading="lazy">, which makes n <a href="Almost_perfect_number" title="Almost perfect number">almost-perfect</a>.

As an example, for two primes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,q:p<q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p,q:p<q}</annotation>

</semantics>

</math><img src="./e9b2e51d1adadd047c3a369e746a23918a770cfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.637ex; height:2.176ex;" alt="{\displaystyle p,q:p<q}" loading="lazy">, let

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n=p\,q}</annotation>

</semantics>

</math><img src="./275ced6393b67b9e7799b7c3288b17bf03325288.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.119ex; height:2.009ex;" alt="{\displaystyle n=p\,q}" loading="lazy">.</dd></dl>

Then

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (n)=(p+1)(q+1)=n+1+(p+q),}</annotation>

</semantics>

</math><img src="./4f8401d8644deae8db5a9e1149ed2e0f15fa835a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.367ex; height:2.843ex;" alt="{\displaystyle \sigma (n)=(p+1)(q+1)=n+1+(p+q),}" loading="lazy"></dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \varphi (n)=(p-1)(q-1)=n+1-(p+q),}</annotation>

</semantics>

</math><img src="./6c10676e66e251ea37d92f8b499a00521eafb437.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.558ex; height:2.843ex;" alt="{\displaystyle \varphi (n)=(p-1)(q-1)=n+1-(p+q),}" loading="lazy"></dd></dl>

and

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n+1=(\sigma (n)+\varphi (n))/2,}</annotation>

</semantics>

</math><img src="./ec80b60b30fe0028aa2041257702ac84d46d7c37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.375ex; height:2.843ex;" alt="{\displaystyle n+1=(\sigma (n)+\varphi (n))/2,}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle p+q=(\sigma (n)-\varphi (n))/2,}</annotation>

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</math><img src="./6e99922a8c5561778359b6e3846288de3990739c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:25.147ex; height:2.843ex;" alt="{\displaystyle p+q=(\sigma (n)-\varphi (n))/2,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (n)}</annotation>

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</math><img src="./f067864064667dd5f8b2508b9cbf983d89788629.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle \varphi (n)}" loading="lazy"> is <a href="Euler_phi" class="mw-redirect" title="Euler phi">Euler's totient function</a>.

Then, the roots of

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<annotation encoding="application/x-tex">{\displaystyle (x-p)(x-q)=x^{2}-(p+q)x+n=x^{2}-[(\sigma (n)-\varphi (n))/2]x+[(\sigma (n)+\varphi (n))/2-1]=0}</annotation>

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</math><img src="./0ad1713c0c16dad5582f05ac235d6f6a116b598a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:90.783ex; height:3.176ex;" alt="{\displaystyle (x-p)(x-q)=x^{2}-(p+q)x+n=x^{2}-[(\sigma (n)-\varphi (n))/2]x+[(\sigma (n)+\varphi (n))/2-1]=0}" loading="lazy"></dd></dl>

express p and q in terms of σ(n) and φ(n) only, requiring no knowledge of n or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+q}">

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<annotation encoding="application/x-tex">{\displaystyle p+q}</annotation>

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</math><img src="./02fcb4cecca0b7e3116a7351e4345b48ef6de371.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.169ex; height:2.343ex;" alt="{\displaystyle p+q}" loading="lazy">, as

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<annotation encoding="application/x-tex">{\displaystyle p=(\sigma (n)-\varphi (n))/4-{\sqrt {[(\sigma (n)-\varphi (n))/4]^{2}-[(\sigma (n)+\varphi (n))/2-1]}},}</annotation>

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</math><img src="./4451d4e5cba4e47140fd055650c0cc07739186a2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; margin-left: -0.089ex; width:69.35ex; height:4.843ex;" alt="{\displaystyle p=(\sigma (n)-\varphi (n))/4-{\sqrt {[(\sigma (n)-\varphi (n))/4]^{2}-[(\sigma (n)+\varphi (n))/2-1]}},}" loading="lazy"></dd>

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<annotation encoding="application/x-tex">{\displaystyle q=(\sigma (n)-\varphi (n))/4+{\sqrt {[(\sigma (n)-\varphi (n))/4]^{2}-[(\sigma (n)+\varphi (n))/2-1]}}.}</annotation>

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</math><img src="./ec0c03afb9b83137b596f0c857be23fa65b66806.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:69.161ex; height:4.843ex;" alt="{\displaystyle q=(\sigma (n)-\varphi (n))/4+{\sqrt {[(\sigma (n)-\varphi (n))/4]^{2}-[(\sigma (n)+\varphi (n))/2-1]}}.}" loading="lazy"></dd></dl>

Also, knowing n and either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (n)}</annotation>

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</math><img src="./c3213bbd075ed40d41d2b72dfaa4a190e031d3fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.534ex; height:2.843ex;" alt="{\displaystyle \sigma (n)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (n)}</annotation>

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</math><img src="./f067864064667dd5f8b2508b9cbf983d89788629.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle \varphi (n)}" loading="lazy">, or, alternatively, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p+q}">

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<annotation encoding="application/x-tex">{\displaystyle p+q}</annotation>

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</math><img src="./02fcb4cecca0b7e3116a7351e4345b48ef6de371.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.169ex; height:2.343ex;" alt="{\displaystyle p+q}" loading="lazy"> and either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \sigma (n)}</annotation>

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</math><img src="./c3213bbd075ed40d41d2b72dfaa4a190e031d3fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.534ex; height:2.843ex;" alt="{\displaystyle \sigma (n)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varphi (n)}">

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<annotation encoding="application/x-tex">{\displaystyle \varphi (n)}</annotation>

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</math><img src="./f067864064667dd5f8b2508b9cbf983d89788629.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.724ex; height:2.843ex;" alt="{\displaystyle \varphi (n)}" loading="lazy"> allows an easy recovery of p and q.

In 1984, <a href="Roger_Heath-Brown" title="Roger Heath-Brown">Roger Heath-Brown</a> proved that the equality

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{0}(n)=\sigma _{0}(n+1)}</annotation>

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</math><img src="./797253e0a1c539d682c5a9d7dcecc6a5b5d8c4d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.273ex; height:2.843ex;" alt="{\displaystyle \sigma _{0}(n)=\sigma _{0}(n+1)}" loading="lazy"></dd></dl>

is true for infinitely many values of n, see <a href="On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="https://oeis.org/A005237" class="extiw external" title="oeis:A005237">A005237</a>.

<div class="mw-heading mw-heading3"><h3 id="Dirichlet_convolutions">Dirichlet convolutions</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Dirichlet_convolution" title="Dirichlet convolution">Dirichlet convolution</a></div>

By definition:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma =\operatorname {Id} *\mathbf {1} }">

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<annotation encoding="application/x-tex">{\displaystyle \sigma =\operatorname {Id} *\mathbf {1} }</annotation>

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</math>By <a href="M%C3%B6bius_inversion_formula" title="Möbius inversion formula">Möbius inversion</a>:<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Id} =\sigma *\mu }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Id} =\sigma *\mu }</annotation>

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</math>

<div class="mw-heading mw-heading2"><h2 id="Series_relations">Series relations</h2></div>

Two <a href="Dirichlet_series" title="Dirichlet series">Dirichlet series</a> involving the divisor function are: <a href="#cite_note-FOOTNOTEHardyWright2008326–328§17.5-10">[10]</a>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}=\zeta (s)\zeta (s-a)\quad {\text{for}}\quad s>1,s>a+1,}</annotation>

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</math><img src="./68901dbe7426b616ed306f891180dfde111de060.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.255ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)}{n^{s}}}=\zeta (s)\zeta (s-a)\quad {\text{for}}\quad s>1,s>a+1,}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta }">

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<annotation encoding="application/x-tex">{\displaystyle \zeta }</annotation>

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</math><img src="./d5c3916703cae7938143d38865f78f27faadd4ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.095ex; height:2.509ex;" alt="{\displaystyle \zeta }" loading="lazy"> is the <a href="Riemann_zeta_function" title="Riemann zeta function">Riemann zeta function</a>. The series for d(n) = σ0(n) gives: <a href="#cite_note-FOOTNOTEHardyWright2008326–328§17.5-10">[10]</a>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {d(n)}{n^{s}}}=\zeta ^{2}(s)\quad {\text{for}}\quad s>1,}</annotation>

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</math><img src="./311f1569393ecb8b5565d5ff6f964aa84578e98a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.598ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {d(n)}{n^{s}}}=\zeta ^{2}(s)\quad {\text{for}}\quad s>1,}" loading="lazy"></dd></dl>

and a <a href="Ramanujan" class="mw-redirect" title="Ramanujan">Ramanujan</a> identity<a href="#cite_note-FOOTNOTEHardyWright2008334–337§17.8-11">[11]</a>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)\sigma _{b}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}},}</annotation>

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</math><img src="./5cc514d035ee8fb340ad1c9a2c9a0a78b662decd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.058ex; height:6.843ex;" alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {\sigma _{a}(n)\sigma _{b}(n)}{n^{s}}}={\frac {\zeta (s)\zeta (s-a)\zeta (s-b)\zeta (s-a-b)}{\zeta (2s-a-b)}},}" loading="lazy"></dd></dl>

which is a special case of the <a href="Rankin%E2%80%93Selberg_method" title="Rankin–Selberg method">Rankin–Selberg convolution</a>.

A <a href="Lambert_series" title="Lambert series">Lambert series</a> involving the divisor function is: <a href="#cite_note-FOOTNOTEHardyWright2008338–341§17.10-12">[12]</a>

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<annotation encoding="application/x-tex">{\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{a}(n)=\sum _{n=1}^{\infty }\sum _{j=1}^{\infty }n^{a}q^{j\,n}=\sum _{n=1}^{\infty }{\frac {n^{a}q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }\operatorname {Li} _{-a}(q^{n})}</annotation>

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</math><img src="./25981aed9a7e662c8338333aee01308c3d5f1af9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:57.52ex; height:7.176ex;" alt="{\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{a}(n)=\sum _{n=1}^{\infty }\sum _{j=1}^{\infty }n^{a}q^{j\,n}=\sum _{n=1}^{\infty }{\frac {n^{a}q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }\operatorname {Li} _{-a}(q^{n})}" loading="lazy"></dd></dl>

for arbitrary <a href="Complex_number" title="Complex number">complex</a> |q| ≤ 1 and a (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Li} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {Li} }</annotation>

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</math><img src="./4b13d1265f1339c746db3220d78abddeedec2f04.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.1ex; height:2.176ex;" alt="{\displaystyle \operatorname {Li} }" loading="lazy"> is the <a href="Polylogarithm" title="Polylogarithm">polylogarithm</a>). This summation also appears as the <a href="Eisenstein_series#Fourier_series" title="Eisenstein series">Fourier series of the Eisenstein series</a> and the <a href="Weierstrass_elliptic_functions" class="mw-redirect" title="Weierstrass elliptic functions">invariants of the Weierstrass elliptic functions</a>.

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k>0}">

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<annotation encoding="application/x-tex">{\displaystyle k>0}</annotation>

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</math><img src="./27b3af208b148139eefc03f0f80fa94c38c5af45.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k>0}" loading="lazy">, there is an explicit series representation with <a href="Ramanujan_sum" class="mw-redirect" title="Ramanujan sum">Ramanujan sums</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{m}(n)}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle c_{m}(n)}</annotation>

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</math><img src="./7a5766de232cfc51177f3e5397412a5cec2ebd6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.886ex; height:2.843ex;" alt="{\displaystyle c_{m}(n)}" loading="lazy"> as :<a href="#cite_note-13">[13]</a>

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{k}(n)=\zeta (k+1)n^{k}\sum _{m=1}^{\infty }{\frac {c_{m}(n)}{m^{k+1}}}.}</annotation>

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</math><img src="./754e6a68cd65c31ee85e9b671660fa6b70e2b4f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.007ex; height:6.843ex;" alt="{\displaystyle \sigma _{k}(n)=\zeta (k+1)n^{k}\sum _{m=1}^{\infty }{\frac {c_{m}(n)}{m^{k+1}}}.}" loading="lazy"></dd></dl>

The computation of the first terms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{m}(n)}">

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<annotation encoding="application/x-tex">{\displaystyle c_{m}(n)}</annotation>

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</math><img src="./7a5766de232cfc51177f3e5397412a5cec2ebd6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.886ex; height:2.843ex;" alt="{\displaystyle c_{m}(n)}" loading="lazy"> shows its oscillations around the "average value" <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \zeta (k+1)n^{k}}">

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<annotation encoding="application/x-tex">{\displaystyle \zeta (k+1)n^{k}}</annotation>

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</math><img src="./a6e016c7996d2725f296c37b5e0072dd5431a3e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.602ex; height:3.176ex;" alt="{\displaystyle \zeta (k+1)n^{k}}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle \sigma _{k}(n)=\zeta (k+1)n^{k}\left[1+{\frac {(-1)^{n}}{2^{k+1}}}+{\frac {2\cos {\frac {2\pi n}{3}}}{3^{k+1}}}+{\frac {2\cos {\frac {\pi n}{2}}}{4^{k+1}}}+\cdots \right]}</annotation>

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</math><img src="./cf656d9d28eb4e6b5948ea58d577e6f17bf3fbdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:62.618ex; height:7.843ex;" alt="{\displaystyle \sigma _{k}(n)=\zeta (k+1)n^{k}\left[1+{\frac {(-1)^{n}}{2^{k+1}}}+{\frac {2\cos {\frac {2\pi n}{3}}}{3^{k+1}}}+{\frac {2\cos {\frac {\pi n}{2}}}{4^{k+1}}}+\cdots \right]}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Growth_rate">Growth rate</h2></div>

In <a href="Big_O_notation#Little-o_notation" title="Big O notation">little-o notation</a>, the divisor function satisfies the inequality:<a href="#cite_note-FOOTNOTEApostol1976296-14">[14]</a><a href="#cite_note-FOOTNOTEHardyWright2008342–347§18.1-15">[15]</a>

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mbox{for all }}\varepsilon >0,\quad d(n)=o(n^{\varepsilon }).}">

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<annotation encoding="application/x-tex">{\displaystyle {\mbox{for all }}\varepsilon >0,\quad d(n)=o(n^{\varepsilon }).}</annotation>

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</math><img src="./3c84f21a5e935c6e1266b8cd557b1c81bd95eedf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.6ex; height:2.843ex;" alt="{\displaystyle {\mbox{for all }}\varepsilon >0,\quad d(n)=o(n^{\varepsilon }).}" loading="lazy"></dd></dl>

More precisely, <a href="Severin_Wigert" class="mw-redirect" title="Severin Wigert">Severin Wigert</a> showed that:<a href="#cite_note-FOOTNOTEHardyWright2008342–347§18.1-15">[15]</a>

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<annotation encoding="application/x-tex">{\displaystyle \limsup _{n\to \infty }{\frac {\log d(n)}{\log n/\log \log n}}=\log 2.}</annotation>

</semantics>

</math><img src="./8f593786269b941e10e27351767203fedf851692.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:31.024ex; height:6.509ex;" alt="{\displaystyle \limsup _{n\to \infty }{\frac {\log d(n)}{\log n/\log \log n}}=\log 2.}" loading="lazy"></dd></dl>

On the other hand, since <a href="Euclid's_theorem" title="Euclid's theorem">there are infinitely many prime numbers</a>,<a href="#cite_note-FOOTNOTEHardyWright2008342–347§18.1-15">[15]</a>

</mrow>

</munder>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \liminf _{n\to \infty }d(n)=2.}</annotation>

</semantics>

</math><img src="./2a5b1fa49c83faf74b736752a7c2d25640551ebd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:15.983ex; height:3.843ex;" alt="{\displaystyle \liminf _{n\to \infty }d(n)=2.}" loading="lazy"></dd></dl>

In <a href="Big-O_notation" class="mw-redirect" title="Big-O notation">Big-O notation</a>, <a href="Peter_Gustav_Lejeune_Dirichlet" title="Peter Gustav Lejeune Dirichlet">Peter Gustav Lejeune Dirichlet</a> showed that the <a href="Average_order_of_an_arithmetic_function" title="Average order of an arithmetic function">average order</a> of the divisor function satisfies the following inequality:<a href="#cite_note-FOOTNOTEApostol1976Theorem_3.3-16">[16]</a><a href="#cite_note-FOOTNOTEHardyWright2008347–350§18.2-17">[17]</a>

</mstyle>

</mrow>

</mrow>

</munder>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mbox{for all }}x\geq 1,\sum _{n\leq x}d(n)=x\log x+(2\gamma -1)x+O({\sqrt {x}}),}</annotation>

</semantics>

</math><img src="./3fc4d704f81cf7e9b0861097fecc2b7f0ded82f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:53.436ex; height:5.676ex;" alt="{\displaystyle {\mbox{for all }}x\geq 1,\sum _{n\leq x}d(n)=x\log x+(2\gamma -1)x+O({\sqrt {x}}),}" loading="lazy"></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy"> is <a href="Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler's gamma constant</a>. Improving the bound <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O({\sqrt {x}})}">

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle O({\sqrt {x}})}</annotation>

</semantics>

</math><img src="./c7f14e35c103373aa2a9f7aaba7a80e279af0207.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.848ex; height:3.009ex;" alt="{\displaystyle O({\sqrt {x}})}" loading="lazy"> in this formula is known as <a href="Divisor_summatory_function#Dirichlet's_divisor_problem" title="Divisor summatory function">Dirichlet's divisor problem</a>.

The behaviour of the sigma function is irregular. The asymptotic growth rate of the sigma function can be expressed by: <a href="#cite_note-FOOTNOTEHardyWright2008469–471§22.9-18">[18]</a>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \limsup _{n\rightarrow \infty }{\frac {\sigma (n)}{n\,\log \log n}}=e^{\gamma },}</annotation>

</semantics>

</math><img src="./b502dcaf2e1913ac184a1ee9bb4346e380b407bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.577ex; height:6.176ex;" alt="{\displaystyle \limsup _{n\rightarrow \infty }{\frac {\sigma (n)}{n\,\log \log n}}=e^{\gamma },}" loading="lazy"></dd></dl>

where lim sup is the <a href="Limit_superior" class="mw-redirect" title="Limit superior">limit superior</a>. This result is <a href="Thomas_Hakon_Gr%C3%B6nwall" title="Thomas Hakon Grönwall">Grönwall</a>'s theorem, published in 1913 (<a href="#CITEREFGrönwall1913">Grönwall 1913</a>). His proof uses <a href="Mertens'_theorems" title="Mertens' theorems">Mertens' third theorem</a>, which says that:

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\to \infty }{\frac {1}{\log n}}\prod _{p\leq n}{\frac {p}{p-1}}=e^{\gamma },}</annotation>

</semantics>

</math><img src="./abe68e9731a801682b802db9a018f24e25ff1b94.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.077ex; height:6.676ex;" alt="{\displaystyle \lim _{n\to \infty }{\frac {1}{\log n}}\prod _{p\leq n}{\frac {p}{p-1}}=e^{\gamma },}" loading="lazy"></dd></dl>

where p denotes a prime.

In 1915, Ramanujan proved that under the assumption of the <a href="Riemann_hypothesis" title="Riemann hypothesis">Riemann hypothesis</a>, Robin's inequality

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ \sigma (n)<e^{\gamma }n\log \log n}</annotation>

</semantics>

</math><img src="./c6e5c5716bf4d40d274c8be7925c17df573b6b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.315ex; height:2.843ex;" alt="{\displaystyle \ \sigma (n)<e^{\gamma }n\log \log n}" loading="lazy"> (where γ is the <a href="Euler%E2%80%93Mascheroni_constant" class="mw-redirect" title="Euler–Mascheroni constant">Euler–Mascheroni constant</a>)</dd></dl>

holds for all sufficiently large n (<a href="#CITEREFRamanujan1997">Ramanujan 1997</a>). The largest known value that violates the inequality is n=<a href="5040_(number)" title="5040 (number)">5040</a>. In 1984, Guy Robin proved that the inequality is true for all n > 5040 <a href="If_and_only_if" title="If and only if">if and only if</a> the Riemann hypothesis is true (<a href="#CITEREFRobin1984">Robin 1984</a>). This is Robin's theorem and the inequality became known after him. Robin furthermore showed that if the Riemann hypothesis is false then there are an infinite number of values of n that violate the inequality, and it is known that the smallest such n > 5040 must be <a href="Superabundant_number" title="Superabundant number">superabundant</a> (<a href="#CITEREFAkbaryFriggstad2009">Akbary & Friggstad 2009</a>). It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for n divisible by the fifth power of a prime (<a href="#CITEREFChoieLichiardopolMoreeSolé2007">Choie et al. 2007</a>).

Robin also proved, unconditionally, that the inequality:

<msup>

</mrow>

</msup>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \ \sigma (n)<e^{\gamma }n\log \log n+{\frac {0.6483\ n}{\log \log n}}}</annotation>

</semantics>

</math><img src="./542854ec433cb3e997225ccc730ea821dd9a1c44.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.426ex; height:5.676ex;" alt="{\displaystyle \ \sigma (n)<e^{\gamma }n\log \log n+{\frac {0.6483\ n}{\log \log n}}}" loading="lazy"></dd></dl>

holds for all n ≥ 3.

A related bound was given by <a href="Jeffrey_Lagarias" title="Jeffrey Lagarias">Jeffrey Lagarias</a> in 2002, who proved that the Riemann hypothesis is equivalent to the statement that:

<msub>

</mrow>

</msub>

<msup>

<msub>

</mrow>

</msub>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma (n)<H_{n}+e^{H_{n}}\log(H_{n})}</annotation>

</semantics>

</math><img src="./ffaab08711111ecd9b929e1c9c9384015d531b3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.586ex; height:3.176ex;" alt="{\displaystyle \sigma (n)<H_{n}+e^{H_{n}}\log(H_{n})}" loading="lazy"></dd></dl>

for every <a href="Natural_number" title="Natural number">natural number</a> n > 1, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H_{n}}</annotation>

</semantics>

</math><img src="./e63458b04288bbe116a9a8037dfae0b36b2c639a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.15ex; height:2.509ex;" alt="{\displaystyle H_{n}}" loading="lazy"> is the nth <a href="Harmonic_number" title="Harmonic number">harmonic number</a>, (<a href="#CITEREFLagarias2002">Lagarias 2002</a>).

<ul><li><a href="Arithmetic_function#Divisor_sum_convolutions" title="Arithmetic function">Divisor sum convolutions</a>, lists a few identities involving the divisor functions</li>

<li><a href="Euler's_totient_function" title="Euler's totient function">Euler's totient function</a>, Euler's phi function</li>

<li><a href="Refactorable_number" title="Refactorable number">Refactorable number</a></li>

<li><a href="Table_of_divisors" title="Table of divisors">Table of divisors</a></li>

<li><a href="Unitary_divisor" title="Unitary divisor">Unitary divisor</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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</style><div class="reflist">

<li id="cite_note-Long_1972_46-1">^ <a href="#cite_ref-Long_1972_46_1-0">a</a> <a href="#cite_ref-Long_1972_46_1-1">b</a> <a href="#CITEREFLong1972">Long (1972</a>, p. 46)

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <a href="#CITEREFPettofrezzoByrkit1970">Pettofrezzo & Byrkit (1970</a>, p. 63)

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFPettofrezzoByrkit1970">Pettofrezzo & Byrkit (1970</a>, p. 58)

</li>

/* start https://en.wikipedia.org/ */

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</style><cite id="CITEREFRamanujan1915" class="citation cs2 cs1-prop-long-vol"><a href="Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Ramanujan, S.</a> (1915), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1433496">"Highly Composite Numbers"</a>, Proceedings of the London Mathematical Society, s2-14 (1): 347–409, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs2_14.1.347">10.1112/plms/s2_14.1.347</a></cite>; see section 47, pp. 405–406, reproduced in Collected Papers of Srinivasa Ramanujan, Cambridge Univ. Press, 2015, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=h1G2CgAAQBAJ&pg=PA124">pp. 124–125</a>

</li>

<li id="cite_note-FOOTNOTEHardyWright2008310_f§16.7-5">^ <a href="#cite_ref-FOOTNOTEHardyWright2008310_f§16.7_5-0">a</a> <a href="#cite_ref-FOOTNOTEHardyWright2008310_f§16.7_5-1">b</a> <a href="#cite_ref-FOOTNOTEHardyWright2008310_f§16.7_5-2">c</a> <a href="#CITEREFHardyWright2008">Hardy & Wright (2008)</a>, pp. 310 f, §16.7.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFEulerBell2004" class="citation arxiv cs1">Euler, Leonhard; Bell, Jordan (2004). "An observation on the sums of divisors". <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0411587">math/0411587</a>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <a rel="nofollow" class="external free" href="https://scholarlycommons.pacific.edu/euler-works/175/">https://scholarlycommons.pacific.edu/euler-works/175/</a>, Découverte d'une loi tout extraordinaire des nombres par rapport à la somme de leurs diviseurs

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> <a rel="nofollow" class="external free" href="https://scholarlycommons.pacific.edu/euler-works/542/">https://scholarlycommons.pacific.edu/euler-works/542/</a>, De mirabilis proprietatibus numerorum pentagonalium

</li>

<li id="cite_note-FOOTNOTEGioiaVaidya1967-9"><a href="#cite_ref-FOOTNOTEGioiaVaidya1967_9-0">^</a> <a href="#CITEREFGioiaVaidya1967">Gioia & Vaidya (1967)</a>.

</li>

<li id="cite_note-FOOTNOTEHardyWright2008326–328§17.5-10">^ <a href="#cite_ref-FOOTNOTEHardyWright2008326–328§17.5_10-0">a</a> <a href="#cite_ref-FOOTNOTEHardyWright2008326–328§17.5_10-1">b</a> <a href="#CITEREFHardyWright2008">Hardy & Wright (2008)</a>, pp. 326–328, §17.5.

</li>

<li id="cite_note-FOOTNOTEHardyWright2008334–337§17.8-11"><a href="#cite_ref-FOOTNOTEHardyWright2008334–337§17.8_11-0">^</a> <a href="#CITEREFHardyWright2008">Hardy & Wright (2008)</a>, pp. 334–337, §17.8.

</li>

<li id="cite_note-FOOTNOTEHardyWright2008338–341§17.10-12"><a href="#cite_ref-FOOTNOTEHardyWright2008338–341§17.10_12-0">^</a> <a href="#CITEREFHardyWright2008">Hardy & Wright (2008)</a>, pp. 338–341, §17.10.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <cite id="CITEREFE._Krätzel1981" class="citation book cs1">E. Krätzel (1981). Zahlentheorie. Berlin: VEB Deutscher Verlag der Wissenschaften. p. 130.</cite> (German)

</li>

<li id="cite_note-FOOTNOTEApostol1976296-14"><a href="#cite_ref-FOOTNOTEApostol1976296_14-0">^</a> <a href="#CITEREFApostol1976">Apostol (1976)</a>, p. 296.

</li>

<li id="cite_note-FOOTNOTEHardyWright2008342–347§18.1-15">^ <a href="#cite_ref-FOOTNOTEHardyWright2008342–347§18.1_15-0">a</a> <a href="#cite_ref-FOOTNOTEHardyWright2008342–347§18.1_15-1">b</a> <a href="#cite_ref-FOOTNOTEHardyWright2008342–347§18.1_15-2">c</a> <a href="#CITEREFHardyWright2008">Hardy & Wright (2008)</a>, pp. 342–347, §18.1.

</li>

<li id="cite_note-FOOTNOTEApostol1976Theorem_3.3-16"><a href="#cite_ref-FOOTNOTEApostol1976Theorem_3.3_16-0">^</a> <a href="#CITEREFApostol1976">Apostol (1976)</a>, Theorem 3.3.

</li>

<li id="cite_note-FOOTNOTEHardyWright2008347–350§18.2-17"><a href="#cite_ref-FOOTNOTEHardyWright2008347–350§18.2_17-0">^</a> <a href="#CITEREFHardyWright2008">Hardy & Wright (2008)</a>, pp. 347–350, §18.2.

</li>

<li id="cite_note-FOOTNOTEHardyWright2008469–471§22.9-18"><a href="#cite_ref-FOOTNOTEHardyWright2008469–471§22.9_18-0">^</a> <a href="#CITEREFHardyWright2008">Hardy & Wright (2008)</a>, pp. 469–471, §22.9.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li><cite id="CITEREFAkbaryFriggstad2009" class="citation cs2">Akbary, Amir; Friggstad, Zachary (2009), <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140411041855/http://webdocs.cs.ualberta.ca/~zacharyf/papers/superabundant.pdf">"Superabundant numbers and the Riemann hypothesis"</a> (PDF), American Mathematical Monthly, 116 (3): 273–275, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2F193009709X470128">10.4169/193009709X470128</a>, archived from <a rel="nofollow" class="external text" href="http://webdocs.cs.ualberta.ca/~zacharyf/Papers/superabundant.pdf">the original</a> (PDF) on 2014-04-11</cite>.</li>

<li><cite id="CITEREFApostol1976" class="citation cs2"><a href="Tom_M._Apostol" title="Tom M. Apostol">Apostol, Tom M.</a> (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90163-3</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0434929">0434929</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0335.10001">0335.10001</a></cite></li>

<li><a href="Eric_Bach" title="Eric Bach">Bach, Eric</a>; <a href="Jeffrey_Shallit" title="Jeffrey Shallit">Shallit, Jeffrey</a>, Algorithmic Number Theory, volume 1, 1996, MIT Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-262-02405-5</bdi>, see page 234 in section 8.8.</li>

<li><cite id="CITEREFCaveneyNicolasSondow2011" class="citation cs2">Caveney, Geoffrey; <a href="Jean-Louis_Nicolas" title="Jean-Louis Nicolas">Nicolas, Jean-Louis</a>; Sondow, Jonathan (2011), <a rel="nofollow" class="external text" href="http://www.integers-ejcnt.org/l33/l33.pdf">"Robin's theorem, primes, and a new elementary reformulation of the Riemann Hypothesis"</a> (PDF), INTEGERS: The Electronic Journal of Combinatorial Number Theory, 11: A33, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1110.5078">1110.5078</a>, <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2011arXiv1110.5078C">2011arXiv1110.5078C</a></cite></li>

<li><cite id="CITEREFChoieLichiardopolMoreeSolé2007" class="citation cs2"><a href="YoungJu_Choie" title="YoungJu Choie">Choie, YoungJu</a>; Lichiardopol, Nicolas; Moree, Pieter; Solé, Patrick (2007), "On Robin's criterion for the Riemann hypothesis", Journal de théorie des nombres de Bordeaux, 19 (2): 357–372, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math.NT/0604314">math.NT/0604314</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.5802%2Fjtnb.591">10.5802/jtnb.591</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1246-7405">1246-7405</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2394891">2394891</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:3207238">3207238</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1163.11059">1163.11059</a></cite></li>

<li><cite id="CITEREFGioiaVaidya1967" class="citation cs2">Gioia, A. A.; Vaidya, A. M. (1967), "Amicable numbers with opposite parity", <a href="The_American_Mathematical_Monthly" title="The American Mathematical Monthly">The American Mathematical Monthly</a>, 74 (8): 969–973, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2315280">10.2307/2315280</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2315280">2315280</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0220659">0220659</a></cite></li>

<li><cite id="CITEREFGrönwall1913" class="citation cs2"><a href="Thomas_Hakon_Gr%C3%B6nwall" title="Thomas Hakon Grönwall">Grönwall, Thomas Hakon</a> (1913), "Some asymptotic expressions in the theory of numbers", Transactions of the American Mathematical Society, 14 (1): 113–122, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2FS0002-9947-1913-1500940-6">10.1090/S0002-9947-1913-1500940-6</a></cite></li>

<li><cite id="CITEREFHardyWright2008" class="citation cs2"><a href="G._H._Hardy" title="G. H. Hardy">Hardy, G. H.</a>; <a href="E._M._Wright" title="E. M. Wright">Wright, E. M.</a> (2008) [1938], An Introduction to the Theory of Numbers, Revised by <a href="Roger_Heath-Brown" title="Roger Heath-Brown">D. R. Heath-Brown</a> and <a href="Joseph_H._Silverman" title="Joseph H. Silverman">J. H. Silverman</a>. Foreword by <a href="Andrew_Wiles" title="Andrew Wiles">Andrew Wiles</a>. (6th ed.), Oxford: <a href="Oxford_University_Press" title="Oxford University Press">Oxford University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-19-921986-5</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2445243">2445243</a>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1159.11001">1159.11001</a></cite></li>

<li><cite id="CITEREFIvić1985" class="citation cs2">Ivić, Aleksandar (1985), The Riemann zeta-function. The theory of the Riemann zeta-function with applications, A Wiley-Interscience Publication, New York etc.: John Wiley & Sons, pp. 385–440, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-471-80634-X</bdi>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:0556.10026">0556.10026</a></cite></li>

<li><cite id="CITEREFLagarias2002" class="citation cs2"><a href="Jeffrey_C._Lagarias" class="mw-redirect" title="Jeffrey C. Lagarias">Lagarias, Jeffrey C.</a> (2002), "An elementary problem equivalent to the Riemann hypothesis", <a href="American_Mathematical_Monthly" class="mw-redirect" title="American Mathematical Monthly">The American Mathematical Monthly</a>, 109 (6): 534–543, <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0008177">math/0008177</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2695443">10.2307/2695443</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0002-9890">0002-9890</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2695443">2695443</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1908008">1908008</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:15884740">15884740</a></cite></li>

<li><cite id="CITEREFLong1972" class="citation cs2">Long, Calvin T. (1972), Elementary Introduction to Number Theory (2nd ed.), Lexington: <a href="D._C._Heath_and_Company" title="D. C. Heath and Company">D. C. Heath and Company</a>, <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/77171950">77171950</a></cite></li>

<li><cite id="CITEREFPettofrezzoByrkit1970" class="citation cs2">Pettofrezzo, Anthony J.; Byrkit, Donald R. (1970), Elements of Number Theory, Englewood Cliffs: <a href="Prentice_Hall" title="Prentice Hall">Prentice Hall</a>, <a href="LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/77081766">77081766</a></cite></li>

<li><cite id="CITEREFRamanujan1997" class="citation cs2"><a href="Srinivasa_Ramanujan" title="Srinivasa Ramanujan">Ramanujan, Srinivasa</a> (1997), "Highly composite numbers, annotated by Jean-Louis Nicolas and Guy Robin", The Ramanujan Journal, 1 (2): 119–153, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FA%3A1009764017495">10.1023/A:1009764017495</a>, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1382-4090">1382-4090</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1606180">1606180</a>, <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:115619659">115619659</a></cite></li>

<li><cite id="CITEREFRobin1984" class="citation cs2">Robin, Guy (1984), "Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann", <a href="Journal_de_Math%C3%A9matiques_Pures_et_Appliqu%C3%A9es" title="Journal de Mathématiques Pures et Appliquées">Journal de Mathématiques Pures et Appliquées</a>, Neuvième Série, 63 (2): 187–213, <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0021-7824">0021-7824</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0774171">0774171</a></cite></li>

<li><cite id="CITEREFWilliams2011" class="citation cs2">Williams, Kenneth S. (2011), Number theory in the spirit of Liouville, London Mathematical Society Student Texts, vol. 76, Cambridge: <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-17562-3</bdi>, <a href="Zbl_(identifier)" class="mw-redirect" title="Zbl (identifier)">Zbl</a> <a rel="nofollow" class="external text" href="https://zbmath.org/?format=complete&q=an:1227.11002">1227.11002</a></cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/DivisorFunction.html">"Divisor Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RobinsTheorem.html">"Robin's Theorem"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="http://mathstat.carleton.ca/~williams/papers/pdf/249.pdf">Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions</a> PDF of a paper by Huard, Ou, Spearman, and Williams. Contains elementary (i.e. not relying on the theory of modular forms) proofs of divisor sum convolutions, formulas for the number of ways of representing a number as a sum of triangular numbers, and related results.</li></ul>

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</style><div id="Divisibility-based_sets_of_integers166" style="font-size:114%;margin:0 4em">Divisibility-based sets of integers</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Overview</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Integer_factorization" title="Integer factorization">Integer factorization</a></li>

<li><a href="Divisor" title="Divisor">Divisor</a></li>

<li><a href="Unitary_divisor" title="Unitary divisor">Unitary divisor</a></li>

<li><a href="Prime_factor" class="mw-redirect" title="Prime factor">Prime factor</a></li>

<li><a href="Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">Fundamental theorem of arithmetic</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="7" style="width:1px;padding:0 0 0 2px"><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Factorization forms</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Prime_number" title="Prime number">Prime</a></li>

<li><a href="Composite_number" title="Composite number">Composite</a></li>

<li><a href="Semiprime" title="Semiprime">Semiprime</a></li>

<li><a href="Pronic_number" title="Pronic number">Pronic</a></li>

<li><a href="Sphenic_number" title="Sphenic number">Sphenic</a></li>

<li><a href="Square-free_integer" title="Square-free integer">Square-free</a></li>

<li><a href="Powerful_number" title="Powerful number">Powerful</a></li>

<li><a href="Perfect_power" title="Perfect power">Perfect power</a></li>

<li><a href="Achilles_number" title="Achilles number">Achilles</a></li>

<li><a href="Smooth_number" title="Smooth number">Smooth</a></li>

<li><a href="Regular_number" title="Regular number">Regular</a></li>

<li><a href="Rough_number" title="Rough number">Rough</a></li>

<li><a href="Unusual_number" title="Unusual number">Unusual</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constrained divisor sums</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Perfect_number" title="Perfect number">Perfect</a></li>

<li><a href="Almost_perfect_number" title="Almost perfect number">Almost perfect</a></li>

<li><a href="Quasiperfect_number" title="Quasiperfect number">Quasiperfect</a></li>

<li><a href="Multiply_perfect_number" title="Multiply perfect number">Multiply perfect</a></li>

<li><a href="Hemiperfect_number" title="Hemiperfect number">Hemiperfect</a></li>

<li><a href="Hyperperfect_number" title="Hyperperfect number">Hyperperfect</a></li>

<li><a href="Superperfect_number" title="Superperfect number">Superperfect</a></li>

<li><a href="Unitary_perfect_number" title="Unitary perfect number">Unitary perfect</a></li>

<li><a href="Semiperfect_number" title="Semiperfect number">Semiperfect</a></li>

<li><a href="Practical_number" title="Practical number">Practical</a></li>

<li><a href="Descartes_number" title="Descartes number">Descartes</a></li>

<li><a href="Erd%C5%91s%E2%80%93Nicolas_number" title="Erdős–Nicolas number">Erdős–Nicolas</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With many divisors</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abundant_number" title="Abundant number">Abundant</a></li>

<li><a href="Primitive_abundant_number" title="Primitive abundant number">Primitive abundant</a></li>

<li><a href="Highly_abundant_number" title="Highly abundant number">Highly abundant</a></li>

<li><a href="Superabundant_number" title="Superabundant number">Superabundant</a></li>

<li><a href="Colossally_abundant_number" title="Colossally abundant number">Colossally abundant</a></li>

<li><a href="Highly_composite_number" title="Highly composite number">Highly composite</a></li>

<li><a href="Superior_highly_composite_number" title="Superior highly composite number">Superior highly composite</a></li>

<li><a href="Weird_number" title="Weird number">Weird</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Aliquot_sequence" title="Aliquot sequence">Aliquot sequence</a>-related</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Untouchable_number" title="Untouchable number">Untouchable</a></li>

<li><a href="Amicable_numbers" title="Amicable numbers">Amicable</a> (<a href="Amicable_triple" title="Amicable triple">Triple</a>)</li>

<li><a href="Sociable_number" title="Sociable number">Sociable</a></li>

<li><a href="Betrothed_numbers" title="Betrothed numbers">Betrothed</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Radix" title="Radix">Base</a>-dependent</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Equidigital_number" title="Equidigital number">Equidigital</a></li>

<li><a href="Extravagant_number" title="Extravagant number">Extravagant</a></li>

<li><a href="Frugal_number" title="Frugal number">Frugal</a></li>

<li><a href="Harshad_number" title="Harshad number">Harshad</a></li>

<li><a href="Polydivisible_number" title="Polydivisible number">Polydivisible</a></li>

<li><a href="Smith_number" title="Smith number">Smith</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other sets</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arithmetic_number" title="Arithmetic number">Arithmetic</a></li>

<li><a href="Deficient_number" title="Deficient number">Deficient</a></li>

<li><a href="Friendly_number" title="Friendly number">Friendly</a></li>

<li><a href="Friendly_number#Solitary_numbers" title="Friendly number">Solitary</a></li>

<li><a href="Sublime_number" title="Sublime number">Sublime</a></li>

<li><a href="Harmonic_divisor_number" title="Harmonic divisor number">Harmonic divisor</a></li>

<li><a href="Refactorable_number" title="Refactorable number">Refactorable</a></li>

<li><a href="Superperfect_number" title="Superperfect number">Superperfect</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-04-30" href="https://en.wikipedia.org/wiki/?title=Divisor_function&oldid=1288116308">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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