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%I #26 Oct 25 2020 03:04:06
%S 1,10,29,83,127,290,345,668,786,1270,1333,2407,2199,3450,3683,5349,
%T 4915,7860,6861,10541,10005,13330,12169,19372,15878,21990,21226,28635,
%U 24391,36830,29793,42798,38657,49150,43815,65238,50655,68610,63771,84836,68923,100050,79509,110639,99822
%N a(n) = Sum_{d|n} sigma_3(d).
%C Inverse Möbius transform applied twice to cubes.
%H Seiichi Manyama, <a href="/A321140/b321140.txt">Table of n, a(n) for n = 1..10000</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: Sum_{k>=1} sigma_3(k)*x^k/(1 - x^k).
%F a(n) = Sum_{d|n} d^3*tau(n/d).
%F From _Jianing Song_, Oct 28 2018: (Start)
%F Multiplicative with a(p^e) = (p^3*(p^(3e+3) - e - 2) + e + 1)/(p^3 - 1)^2.
%F Dirichlet g.f.: zeta(s)^2*zeta(s-3). (End)
%F Sum_{k=1..n} a(k) ~ Pi^8 * n^4 / 32400. - _Vaclav Kotesovec_, Nov 08 2018
%p with(numtheory): seq(coeff(series(add(sigma[3](k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 45); # _Muniru A Asiru_, Oct 28 2018
%t Table[Sum[DivisorSigma[3, d], {d, Divisors[n]}] , {n, 45}]
%t nmax = 45; Rest[CoefficientList[Series[Sum[DivisorSigma[3, k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t f[p_, e_] := (p^3*(p^(3e+3) - e - 2) + e + 1)/(p^3 - 1)^2; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 25 2020 *)
%o (PARI) a(n) = sumdiv(n, d, sigma(d, 3)); \\ _Michel Marcus_, Oct 28 2018
%Y Cf. A001158, A007429, A007433, A027848.
%K nonn,mult,easy
%O 1,2
%A _Ilya Gutkovskiy_, Oct 28 2018
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