A288571 a(n) = Sum_{d|n} (-1)^(n/d+1)*tau(d), where tau = number of divisors (A000005).

1, 1, 3, 0, 3, 3, 3, -2, 6, 3, 3, 0, 3, 3, 9, -5, 3, 6, 3, 0, 9, 3, 3, -6, 6, 3, 10, 0, 3, 9, 3, -9, 9, 3, 9, 0, 3, 3, 9, -6, 3, 9, 3, 0, 18, 3, 3, -15, 6, 6, 9, 0, 3, 10, 9, -6, 9, 3, 3, 0, 3, 3, 18, -14, 9, 9, 3, 0, 9, 9, 3, -12, 3, 3, 18, 0, 9, 9, 3, -15, 15, 3, 3, 0, 9

Offset

1,3

Comments

Dirichlet convolution of A048272 and A000012. - Vaclav Kotesovec, Jan 13 2024

Links

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

Formula

G.f.: Sum_{k>=1} tau(k)*x^k/(1 + x^k).

L.g.f.: log(Product_{k>=1} (1 + x^k)^(tau(k)/k)) = Sum_{n>=1} a(n)*x^n/n.

Multiplicative with a(2^e) = (e+1)*(2-e)/2, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - Amiram Eldar, Oct 25 2020

From Amiram Eldar, Sep 14 2023: (Start)

Dirichlet g.f.: (1- 1/2^(s-1)) * zeta(s)^3.

Sum_{k=1..n} a(k) ~ log(2) * n * (log(n) + 3*gamma - 1 - log(2)/2), where gamma is Euler's constant (A001620). (End)

Maple

with(numtheory): seq(add((-1)^(n/a+1)*tau(a), a=divisors(n)), n=1..85); # Paolo P. Lava, Aug 24 2018

Mathematica

Table[Sum[(-1)^(n/d + 1) DivisorSigma[0, d], {d, Divisors[n]}], {n, 85}]
nmax = 85; Rest[CoefficientList[Series[Sum[DivisorSigma[0, k] x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
nmax = 85; Rest[CoefficientList[Series[Log[Product[(1 + x^k)^(DivisorSigma[0, k]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
f[p_, e_] := If[p == 2, (e + 1)*(2 - e)/2, (e + 1)*(e + 2)/2]; 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)*numdiv(d)); \\ Michel Marcus, Aug 24 2018

Crossrefs

Cf. A000005, A001620, A007425, A017113 (positions of 0's), A048272, A288417, A317531.

Sequence in context: A066958 A357062 A066851 * A270028 A167223 A341480

Adjacent sequences: A288568 A288569 A288570 * A288572 A288573 A288574

Keyword

sign,mult,hear,easy

Author

Ilya Gutkovskiy, Aug 23 2018

Status

approved