

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
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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), 162 (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^21), 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] (* JeanFrançois 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



