A078306 - OEIS

%I #40 Aug 07 2022 07:01:55

%S 1,3,10,11,26,30,50,43,91,78,122,110,170,150,260,171,290,273,362,286,

%T 500,366,530,430,651,510,820,550,842,780,962,683,1220,870,1300,1001,

%U 1370,1086,1700,1118,1682,1500,1850,1342,2366,1590,2210,1710,2451,1953

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

%H Seiichi Manyama, <a href="/A078306/b078306.txt">Table of n, a(n) for n = 1..10000</a>

%H J. W. L. Glaisher, <a href="https://books.google.com/books?id=bLs9AQAAMAAJ&amp;pg=RA1-PA1">On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares</a>, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).

%H Heekyoung Hahn, <a href="http://arxiv.org/abs/1507.04426">Convolution sums of some functions on divisors</a>, arXiv:1507.04426 [math.NT], 2015.

%H <a href="/index/Ge#Glaisher">Index entries for sequences mentioned by Glaisher</a>

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

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

%F 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

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

%F From _Vaclav Kotesovec_, Aug 07 2022: (Start)

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

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

%t a[n_] := Sum[(-1)^(n/d+1)*d^2, {d, Divisors[n]}]; Array[a, 50] (* _Jean-François Alcover_, Apr 17 2014 *)

%t 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 *)

%o (PARI) a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^2); \\ _Michel Marcus_, Jul 06 2016

%o (Python)

%o from sympy import divisors

%o print([sum((-1)**(n//d + 1)*d**2 for d in divisors(n)) for n in range(1, 51)]) # _Indranil Ghosh_, Apr 05 2017

%Y Cf. A000593, A064027, A026007, A001157.

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

%K mult,nonn,easy

%O 1,2

%A _Vladeta Jovovic_, Nov 22 2002