A283271 Expansion of exp( Sum_{n>=1} -sigma_5(n)*x^n/n ) in powers of x.

1, -1, -16, -65, -55, 807, 4809, 13135, 550, -169070, -862710, -2281174, -1221309, 20194565, 114391575, 346400092, 486546751, -1239516671, -11089537215, -41702958960, -93143227027, -45337210750, 674845109986, 3682196642725, 11405949184465, 20796945542222

Offset: 0

Author

Seiichi Manyama, Mar 04 2017

Keywords

• sign

Comments

Let A(x) denote the g.f. and let m be an integer. Define a sequence by u(n) = [x^n] A(x)^(m*n). We conjecture that the supercongruence u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) holds for all positive integers n and r and all primes p >= 7. Cf. A380581. - Peter Bala, Jan 21 2025

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=4 of A283272.

Cf. A023873 (exp( Sum_{n>=1} sigma_5(n)*x^n/n )).

Cf. exp( Sum_{n>=1} -sigma_k(n)*x^n/n ): A010815 (k=1), A073592 (k=2), A283263 (k=3), A283264 (k=4), this sequence (k=5).

Formula

G.f.: Product_{n>=1} (1 - x^n)^(n^4).

a(n) = -(1/n)*Sum_{k=1..n} sigma_5(k)*a(n-k).