A078306 a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^2.

1, 3, 10, 11, 26, 30, 50, 43, 91, 78, 122, 110, 170, 150, 260, 171, 290, 273, 362, 286, 500, 366, 530, 430, 651, 510, 820, 550, 842, 780, 962, 683, 1220, 870, 1300, 1001, 1370, 1086, 1700, 1118, 1682, 1500, 1850, 1342, 2366, 1590, 2210, 1710, 2451, 1953

Offset: 1

Author

Vladeta Jovovic, Nov 22 2002

Keywords

• mult
• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
• Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
• Index entries for sequences mentioned by Glaisher

Crossrefs

Cf. A000593, A064027, A026007, A001157.

Glaisher's zeta'_i (i=0..12): A048272, A000593, A078306, A078307, A284900, A284926, A284927, A321552, A321553, A321554, A321555, A321556, A321557

Programs

Mathematica

a[n_] := Sum[(-1)^(n/d+1)*d^2, {d, Divisors[n]}]; Array[a, 50] (* Jean-François Alcover, Apr 17 2014 *)
Table[CoefficientList[Series[-Log[Product[1/(x^k + 1)^k, {k, 1, 90}]], {x, 0, 80}], x][[n + 1]] n, {n, 1, 80}] (* Benedict W. J. Irwin, Jul 05 2016 *)

PARI

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^2); \\ Michel Marcus, Jul 06 2016

Python

from sympy import divisors
print([sum((-1)**(n//d + 1)*d**2 for d in divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Apr 05 2017

Formula

G.f.: Sum_{n >= 1} n^2*x^n/(1+x^n).

Multiplicative with a(2^e) = (2*4^e+1)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.

L.g.f.: -log(Product_{ k>0 } 1/(x^k+1)^k) = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016

G.f.: Sum_{n >= 1} (-1)^(n+1) * x^n*(1 + x^n)/(1 - x^n)^3. - Peter Bala, Jan 14 2021

From Vaclav Kotesovec, Aug 07 2022: (Start)

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

Sum_{k=1..n} a(k) ~ zeta(3) * n^3 / 4. (End)