Divisibility sequence ($m\mid n⟹a_m\mid a_n$)

A divisibility sequence is an integer sequence satisfying $m\mid n⟹a_m\mid a_n$, with any conventions about zero terms stated. For example $a_n=2^n-1$ has this property because if $n=km$, thenThe 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.

A divisibility sequence associated with multiples of a point on an elliptic curve, using suitably normalized division polynomials. With $W_0=0$ and $W_1=1$, its terms satisfy the Ward identity $W_{m+n}{W}_{m-n}=W_{m+1}{W}_{m-1}{W}_n^2-W_{n+1}{W}_{n-1}{W}_m^2$. Setting nearby indices gives doubling identities, enabling logarithmic-depth evaluation.