Divisibility sequence

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 strong 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 = (1, 1, 2, 3, 5, 8,...) form a strong divisibility sequence.
More generally, Lucas sequences of the first kind are divisibility sequences.
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 978-0-8218-3387-2.
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 (4): 334–336. doi:10.1090/s0002-9904-1939-06980-2.
Hoggatt, Jr., V. E.; Long, C. T. (1973). "Divisibility properties of generalized fibonacci polynomials" (PDF). Fibonacci Quarterly: 113.
Bézivin, J.-P.; Pethö, 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; Dinakar Ramakrishnan; Kenneth A. Ribet; John Tate, Number Theory, Analysis and Geometry. In Memory of Serge Lang, Springer, pp. 243–271, ISBN 978-1-4614-1259-5

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