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A078306 a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^2. 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

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

MATHEMATICA

a[n_] := Sum[(-1)^(n/d+1)*d^2, {d, Divisors[n]}]; Array[a, 50] (* Jean-Fran├žois 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 *)

PROG

(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

CROSSREFS

Cf. A000593, A064027, A026007.

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

Sequence in context: A169939 A073108 A255160 * A136815 A119223 A119165

Adjacent sequences:  A078303 A078304 A078305 * A078307 A078308 A078309

KEYWORD

mult,nonn,changed

AUTHOR

Vladeta Jovovic, Nov 22 2002

STATUS

approved

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Last modified December 9 03:27 EST 2019. Contains 329872 sequences. (Running on oeis4.)