A078307 - OEIS

%I #41 Oct 27 2022 07:32:08

%S 1,7,28,55,126,196,344,439,757,882,1332,1540,2198,2408,3528,3511,4914,

%T 5299,6860,6930,9632,9324,12168,12292,15751,15386,20440,18920,24390,

%U 24696,29792,28087,37296,34398,43344,41635,50654,48020,61544,55314,68922,67424

%N a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^3.

%H Seiichi Manyama, <a href="/A078307/b078307.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Alois P. Heinz)

%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^3*x^n/(1+x^n).

%F Multiplicative with a(2^e) = (6*8^e+1)/7, a(p^e) = (p^(3*e+3)-1)/(p^3-1), p > 2.

%F 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

%F Sum_{k=1..n} a(k) ~ c * n^4, where c = 7*Pi^4/2880 = 0.236758... . - _Amiram Eldar_, Oct 27 2022

%p with(numtheory):

%p a:= n-> add((-1)^(n/d+1)*d^3, d=divisors(n)):

%p seq(a(n), n=1..70);  # _Alois P. Heinz_, Aug 03 2013

%t a[n_] := Sum[(-1)^(n/d+1)*d^3, {d, Divisors[n]}]; Table[a[n], {n, 1, 70}] (* _Jean-François Alcover_, Jan 17 2014 *)

%t 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 *)

%o (PARI) a(n) = sumdiv(n, d, (-1)^(n/d + 1)*d^3); \\ _Indranil Ghosh_, Apr 05 2017

%o (Python)

%o from sympy import divisors

%o print([sum((-1)**(n//d + 1)*d**3 for d in divisors(n)) for n in range(1, 51)]) # _Indranil Ghosh_, Apr 05 2017

%Y Cf. A000593, A078306, A027998.

%K mult,nonn

%O 1,2

%A _Vladeta Jovovic_, Nov 22 2002