A078307 a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^3.
1, 7, 28, 55, 126, 196, 344, 439, 757, 882, 1332, 1540, 2198, 2408, 3528, 3511, 4914, 5299, 6860, 6930, 9632, 9324, 12168, 12292, 15751, 15386, 20440, 18920, 24390, 24696, 29792, 28087, 37296, 34398, 43344, 41635, 50654, 48020, 61544, 55314, 68922, 67424
Offset: 1
Links
- Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)
- 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.
Programs
-
Maple
with(numtheory): a:= n-> add((-1)^(n/d+1)*d^3, d=divisors(n)): seq(a(n), n=1..70); # Alois P. Heinz, Aug 03 2013
-
Mathematica
a[n_] := Sum[(-1)^(n/d+1)*d^3, {d, Divisors[n]}]; Table[a[n], {n, 1, 70}] (* Jean-François Alcover, Jan 17 2014 *) f[p_, e_] := (p^(3*e + 3) - 1)/(p^3 - 1); f[2, e_] := (6*8^e + 1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 42] (* Amiram Eldar, Oct 27 2022 *) -
PARI
a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^3); \\ Indranil Ghosh, Apr 05 2017
-
Python
from sympy import divisors print([sum((-1)**(n//d + 1)*d**3 for d in divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Apr 05 2017
Formula
G.f.: Sum_{n >= 1} n^3*x^n/(1+x^n).
Multiplicative with a(2^e) = (6*8^e+1)/7, a(p^e) = (p^(3*e+3)-1)/(p^3-1), p > 2.
L.g.f.: log(Product_{k>=1} (1 + x^k)^(k^2)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Mar 12 2018
Sum_{k=1..n} a(k) ~ c * n^4, where c = 7*Pi^4/2880 = 0.236758... . - Amiram Eldar, Oct 27 2022