NJAS> Here is the preface to a file:
This is a list, in OEIS numerical order, of divisibility sequences.
I confess to having looked up only a few of the 60-odd sequences, but it may come as news to some that there are elliptic divisibility sequences of polynomials. E.g. s[1]:s[2]:1,s[3]:-1,s[4]:x,s[n+4] :=(s[n+3]*s[n+1] + s[n+2]^2)/s[n] gives 1 1 2 1 3 - 1 4 x 5 x + 1 2 6 x - x - 1 3 7 - (x + x + 1) 8 - x (3 x + 2) 5 4 2 9 x - x + 3 x + 3 x + 1 5 4 3 2 10 - (x + 1) (x - x + 3 x + 2 x - 2 x - 1) 7 6 5 4 3 2 11 - (x - 2 x + 3 x + 9 x + 5 x + 3 x + 3 x + 1) 2 6 4 3 2 12 - x (x - x - 1) (x + 2 x + 5 x + 9 x + 9 x + 3) . . . where s[<prime>] is irreducible, etc. --rwg