Divisibility sequence

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

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

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 mn,

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

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.