Divisibility sequence

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

${\displaystyle {\text{if }}m\mid n{\text{ then }}a_{m}\mid a_{n}}$

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 ${\displaystyle (a_{n})}$ such that for all positive integers m, n,

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

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