= Divisibility sequence
{title2=$m\mid n\Longrightarrow a_m\mid a_n$}
A <divisibility sequence> is an <integer sequence> satisfying $m\mid n\Longrightarrow a_m\mid a_n$, with any conventions about zero terms stated. For example $a_n=2^n-1$ has this property because if $n=km$, then
$$
2^{km}-1=(2^m-1)(1+2^m+\cdots+2^{(k-1)m}).
$$
The <Fibonacci numbers> also form a <divisibility sequence>. The stronger identity $\gcd(a_m,a_n)=|a_{\gcd(m,n)}|$ defines a strong divisibility property and does not follow from the basic implication alone. Such sequences organize common-factor questions by relationships among their indices.
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