A327095 Expansion of Sum_{k>=1} k * x^k * (1 - x^k + x^(2*k)) / (1 - x^(4*k)).

1, 1, 4, 2, 6, 4, 8, 4, 13, 6, 12, 8, 14, 8, 24, 8, 18, 13, 20, 12, 32, 12, 24, 16, 31, 14, 40, 16, 30, 24, 32, 16, 48, 18, 48, 26, 38, 20, 56, 24, 42, 32, 44, 24, 78, 24, 48, 32, 57, 31, 72, 28, 54, 40, 72, 32, 80, 30, 60, 48, 62, 32, 104, 32, 84, 48

Offset: 1

Author

Ilya Gutkovskiy, Sep 13 2019

Keywords

• nonn
• mult

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002131, A115607, A285895, A316631 (Moebius transform).

Programs

Maple

N:= 100:
G:= add(k * x^k * (1 - x^k + x^(2*k)) / (1 - x^(4*k)),k=1..N):
S:= series(G,x,N+1):
[seq(coeff(S,x,i),i=1..N)];# Robert Israel, Sep 17 2019

Mathematica

nmax = 66; CoefficientList[Series[Sum[k x^k (1 - x^k + x^(2 k))/(1 - x^(4 k)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
A002131[n_] := Total[Select[Divisors[n], OddQ[n/#] &]]; a[n_] := If[OddQ[n], A002131[n], A002131[n] - A002131[n/2]]; Table[a[n], {n, 1, 66}]
f[p_, e_] := (p^(e + 1) - 1)/(p - 1); f[2, e_] := 2^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

PARI

a(n)={sumdiv(n, d, d*((n/d%2==1) - (n/d%4==2)))} \\ Andrew Howroyd, Sep 13 2019

Formula

G.f.: Sum_{k>=1} x^k * (1 + 3 * x^(2*k) + x^(3*k) + 3 * x^(4*k) + x^(6*k)) / (1 - x^(4*k))^2.

a(n) = Sum_{d|n, n/d odd} d - Sum_{d|n, n/d twice odd} d.

a(n) = A002131(n) if n odd, A002131(n) - A002131(n/2) if n even.

From Amiram Eldar, Dec 05 2022: (Start)

Multiplicative with a(2^e) = 2^(e-1), and a(p^e) = (p^(e+1)-1)/(p-1) if p > 2.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/64 = 0.462637... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/2^s)^2. - Amiram Eldar, Jan 06 2023