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In mathematics, a divisibility sequence is an integer sequence {(a_n)}_{n\in\N} such that for all natural numbers m, n,

$$\text{if }m\mid n\text{ then }a_m\mid a_n,$$

i.e., 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)}_{n\in\N} such that for all natural numbers m, n,

$$\gcd(a_m,a_n) = a_{\gcd(m,n)}.$$

Note that a strong divisibility sequence is immediately a divisibility sequence; if m\mid n, immediately gcd(m,n) = m. Then by the strong divisibility property, $$gcd(a_m,a_n) = a_m$$ and therefore $$a_m\mid a_n$$.

Examples

Any constant sequence is a divisibility sequence.
Every sequence of the form $$a_n = kn$$, for some nonzero integer k, is a divisibility sequence.
Every sequence of the form $$a_n = A^n - B^n$$ for integers A>B>0 is a divisibility sequence.
The Fibonacci numbers F = (0, 1, 1, 2, 3, 5, 8,...) form a strong divisibility sequence.
Elliptic divisibility sequences are another class of such sequences.

References

Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence Sequences. American Mathematical Society. ISBN 978-0821833872.
Hall, Marshall (1936). "Divisibility sequences of third order". Am. J. Math 58: 577–584. JSTOR 2370976.
Ward, Morgan (1939). "A note on divisibility sequences". Bull. Amer. Math. Soc 45: 334–336.
Hoggat, Jr., V. E.; Long, C. T. (1973). "Divisibility properties of generalized fibonacci polynomials". Fibonacci Quarterly: 113.
Bézivin, J.-P.; Ethö, A.; van der Porten, A. J. (1990). "A full characterization of divisibility sequences". Am. J. Math. 112 (6): 985–1001. JSTOR 2374733.
P. Ingram; J. H. Silverman (2012), "Primitive divisors in elliptic divisibility sequences", in Dorian Goldfeld; Jay Jorgenson; Peter Jones et al., Number Theory, Analysis and Geometry. In Memory of Serge Lang, Springer, pp. 243-271, ISBN 978-1-4614-1259-5

Some divisibility sequences listed in the On-line Encyclopedia of Integer Sequences.