A320416 G.f.: exp( Sum_{n>=1} sigma_2(n)*sigma_3(n)/sigma(n) * x^n / n ), where sigma_{k}(n) equals the sum of the k-th powers of the divisors of n.

1, 1, 8, 31, 110, 380, 1258, 4145, 13062, 40549, 123177, 367524, 1078214, 3117641, 8889005, 25019907, 69592393, 191419803, 521053616, 1404437257, 3750615009, 9928671171, 26065603448, 67890853023, 175503159857, 450440887668, 1148173347232, 2907505610383, 7316343339760, 18299391020158, 45503950232785, 112518903733379

Offset

0,3

Comments

This sequence is conjectured to consist entirely of integers.

Related to A320917(n) = sigma_2(n)*sigma_3(n)/sigma(n).

Links

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Example

G.f.: A(x) = 1 + x + 8*x^2 + 31*x^3 + 110*x^4 + 380*x^5 + 1258*x^6 + 4145*x^7 + 13062*x^8 + 40549*x^9 + 123177*x^10 + ...
such that
log(A(x)) = x + 15*x^2/2 + 70*x^3/3 + 219*x^4/4 + 546*x^5/5 + 1050*x^6/6 + 2150*x^7/7 + 3315*x^8/8 + 5299*x^9/9 + 8190*x^10/10 + 13542*x^11/11 + 15330*x^12/12 + ... + sigma_2(n)*sigma_3(n)/sigma(n)*x^n/n + ...

Mathematica

nmax = 30; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k]*DivisorSigma[3, k]/DivisorSigma[1, k] * x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 06 2020 *)

Prog

(PARI) {a(n) = polcoeff(exp( sum(m=1, n, sigma(m, 2)*sigma(m, 3)/sigma(m) *x^m/m ) +x*O(x^n)), n)}
for(n=0, 35, print1(a(n), ", "))

Crossrefs

Cf. A320917.

Sequence in context: A212064 A213764 A335606 * A289613 A055845 A034556

Adjacent sequences: A320413 A320414 A320415 * A320417 A320418 A320419

Keyword

nonn

Author

Paul D. Hanna, Oct 26 2018

Status

approved