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In mathematics, a divisibility sequence is an integer sequence ( a n ) {\displaystyle (a_{n})} indexed by positive integers n such that

if m ∣ ∣ n then a m ∣ ∣ a n {\displaystyle {\text{if }}m\mid n{\text{ then }}a_{m}\mid a_{n}}

for all m, n. That is, whenever one index is a multiple of another one, then the corresponding term also is a multiple of the other term. The concept can be generalized to sequences with values in any ring where the concept of divisibility is defined.

A strong divisibility sequence is an integer sequence ( a n ) {\displaystyle (a_{n})} such that for all positive integers m, n,

gcd ( a m , a n ) = a gcd ( m , n ) . {\displaystyle \gcd(a_{m},a_{n})=a_{\gcd(m,n)}.}

Every strong divisibility sequence is a divisibility sequence: gcd ( m , n ) = m {\displaystyle \gcd(m,n)=m} if and only if m ∣ ∣ n {\displaystyle m\mid n} . Therefore, by the strong divisibility property, gcd ( a m , a n ) = a m {\displaystyle \gcd(a_{m},a_{n})=a_{m}} and therefore a m ∣ ∣ a n {\displaystyle a_{m}\mid a_{n}} .

• Examples

• References

• Any constant sequence is a strong divisibility sequence.

• Every sequence of the form a n = k n , {\displaystyle a_{n}=kn,} for some nonzero integer k, is a divisibility sequence.

• The numbers of the form 2 n − − 1 {\displaystyle 2^{n}-1} (Mersenne numbers) form a strong divisibility sequence.

• The repunit numbers in any base Rn(b) form a strong divisibility sequence.

• More generally, any sequence of the form a n = A n − − B n {\displaystyle a_{n}=A^{n}-B^{n}} for integers A > B > 0 {\displaystyle A>B>0} is a divisibility sequence. In fact, if A {\displaystyle A} and B {\displaystyle B} are coprime, then this is a strong divisibility sequence.

• The Fibonacci numbers Fn form a strong divisibility sequence.

• More generally, any Lucas sequence of the first kind Un(P,Q) is a divisibility sequence. Moreover, it is a strong divisibility sequence when gcd(P,Q) = 1.

• Elliptic divisibility sequences are another class of such sequences.

• citerefeverestvan-der-poortenshparlinskiward2003Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence Sequences. American Mathematical Society. ISBN 978-0-8218-3387-2.

• citerefhall1936Hall, Marshall (1936). "Divisibility sequences of third order". Am. J. Math. 58 (3): 577–584. doi:10.2307/2370976. JSTOR 2370976.

• citerefward1939Ward, Morgan (1939). "A note on divisibility sequences". Bull. Amer. Math. Soc. 45 (4): 334–336. doi:10.1090/s0002-9904-1939-06980-2.

• citerefhoggatt-jr-long1973Hoggatt, Jr., V. E.; Long, C. T. (1973). "Divisibility properties of generalized Fibonacci polynomials" (PDF). Fibonacci Quarterly: 113.

• citerefb-zivinpeth-van-der-porten1990Bézivin, J.-P.; Pethö, A.; van der Porten, A. J. (1990). "A full characterization of divisibility sequences". Am. J. Math. 112 (6): 985–1001. doi:10.2307/2374733. JSTOR 2374733.

• citerefp-ingramj-h-silverman2012P. Ingram; J. H. Silverman (2012), "Primitive divisors in elliptic divisibility sequences", in Dorian Goldfeld; Jay Jorgenson; Peter Jones; Dinakar Ramakrishnan; Kenneth A. Ribet; John Tate (eds.), Number Theory, Analysis and Geometry. In Memory of Serge Lang, Springer, pp. 243–271, ISBN 978-1-4614-1259-5