A318768 a(n) = Sum_{d|n} (-1)^(n/d+1) * Sum_{j|d} tau(j), where tau = number of divisors (A000005).

1, 2, 4, 2, 4, 8, 4, 0, 10, 8, 4, 8, 4, 8, 16, -5, 4, 20, 4, 8, 16, 8, 4, 0, 10, 8, 20, 8, 4, 32, 4, -14, 16, 8, 16, 20, 4, 8, 16, 0, 4, 32, 4, 8, 40, 8, 4, -20, 10, 20, 16, 8, 4, 40, 16, 0, 16, 8, 4, 32, 4, 8, 40, -28, 16, 32, 4, 8, 16, 32, 4, 0, 4, 8, 40, 8, 16, 32, 4

Offset: 1

Author

Ilya Gutkovskiy, Sep 03 2018

Keywords

• sign
• mult
• easy

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000005, A007425, A007426, A051062 (positions of 0's), A288571.

Cf. A001620, A082633.

Programs

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 *)

PARI

a(n) = sumdiv(n, d, (-1)^(n/d+1) * sumdiv(d, j, numdiv(j))); \\ Michel Marcus, Sep 04 2018

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)).

Sum_{k=1..n} a(k) ~ (log(2)/2) * n * (log(n)^2 + (8 * gamma - log(2) - 2) * log(n) + 12 * gamma^2 - 8 * gamma + log(2) + 2 - 4 * gamma * log(2) + log(2)^2/3 - 8 * gamma_1), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633). (End)