What is "divisibility sequence"

n. (context mathematics English) A sequence of integers having the property that, if their respective positions in the sequence divide, then their values divide

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

if m ∣ nthen a ∣ a, 

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, a) = a.

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