a sequence [of numbers] is called a divisibility sequence, when
if m|n then a(m)|a(n)
If the converse is true then we call it a "strict divisibility sequence"

since 1|n then any a(n) is divisible by a(1)
for normalization we assume a(1)=1
    
any number divides 0
so any a(n) must divide a(0)
so a(0)=0


[ a(-n) ?]

We are interested in Fermat's idea:
considering the subsequence b(m)=a(2^m)
which is a chain of divisors a(1)|a(2)|a(4)|a(8)|...
or b(0)|b(1)|b(2)|b(3)|...
and we define f(m)=b(m+1)/b(m)


trivial example
a(n)=1
f(m)=1


natural/identity example
a(n)=n
f(m)=2


factorial
a(n)=n!


[?] a(0)=0!=1




Lucas: generalized Mersenne/Fibonacci

a(n)=(a^n-b^n)/(a-b)
f(m)=a^2^m+b^2^m