It may be that someone else is interested in (parts of) this reply that I just made. Can anyone prove that A047817 in OEIS is in fact a divisibility sequence? R. ---------- Forwarded message ---------- Date: Tue, 15 Dec 2009 13:57:08 -0700 (MST) From: Richard Guy <rkg@cpsc.ucalgary.ca> To: Georgi Guninski <guninski@guninski.com> Cc: Richard Guy <rkg@cpsc.ucalgary.ca> Subject: Re: [seqfan] Divisibility sequences in OEIS. The Hurwitz numbers evidently form a div seq, but I don't know what they are, so don't know how to prove it. [snip] More exciting is that the numbers of points (including the pt at infty) on any elliptic curve over the field F(q^n) form a fourth order divisibility sequence, with nth term q^n + 1 - t_n where {t_n} is a 2nd order recurring sequence generated by x^2 - tx + q, with t_0 = 2 (giving a(0) = q^0 + 1 - 2 = 0) and t_1 = t where t is the `trace of Frobenius' (Hecke eigenvalue) for the curve mod q. a(1) = q+1-t is a common factor of all the terms. It's an interesting (at least to me) paradox that 2 + 2 = 2 x 2. I.e., the sum of two second order recurring sequences [ q^n + 1 satisfies the recurrence x^2 - (q+1)x + q ] form a 4th order one, which is the product of 2 second order div seqs (of integers in a quadratic extension field of Q), the 4 roots being the products of the 2 roots of each of the two quadratics. There's a 2-parameter (q \& t) infinite family of such sequences, but A002248 is the only one I've spotted in OEIS. Your search didn't pick this one up. Several hundred should be in OEIS, e.g., for q = the first 8 or 10 primes and t = 0, +-1, +-2, ... Note that although Hasse tells us that |t| <= 2 * sqrt(q), you still get divisibility seqs with t outside that range. The fact that it's a divisibility sequence follows from the group structure of points on an elliptic curve. Also my colleague Hugh Williams has recently proved that a large (3-parameter) family of 4th order recurring sequences, which includes these as a particular case, are div seqs. [snip]