OFFSET
1,2
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
FORMULA
G.f.: Sum_{k>=1} tau_3(k)*x^k/(1 + x^k), where tau_3() = A007425.
L.g.f.: log(Product_{k>=1} (1 + x^k)^(tau_3(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
Multiplicative with a(2^e) = 1 + (7-e^2)*e/6, and a(p^e) = binomial(e+3,3) for an odd prime p. - Amiram Eldar, Oct 25 2020
From Amiram Eldar, Dec 18 2023: (Start)
Dirichlet g.f.: zeta(s)^4 * (1 - 1/2^(s-1)).
MATHEMATICA
Table[Sum[(-1)^(n/d + 1) Sum[DivisorSigma[0, j], {j, Divisors[d]}], {d, Divisors[n]}], {n, 79}]
nmax = 79; Rest[CoefficientList[Series[Sum[DivisorSum[k, DivisorSigma[0, #] &] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 79; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSum[k, DivisorSigma[0, #] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := If[p == 2, 1 + (7 - e^2)*e/6, Binomial[e + 3, 3]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *)
PROG
(PARI) a(n) = sumdiv(n, d, (-1)^(n/d+1) * sumdiv(d, j, numdiv(j))); \\ Michel Marcus, Sep 04 2018
CROSSREFS
KEYWORD
sign,mult,easy
AUTHOR
Ilya Gutkovskiy, Sep 03 2018
STATUS
approved