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a(n) ~ exp(3*((Pi^2 - 6)*Zeta(3))^(1/3) * n^(2/3) / (2*Pi)^(2/3) + 1/4) * ((Pi^2 - 6)*Zeta(3))^(1/4) / (A^3 * 2^(1/12) * 3^(1/2) * Pi^(5/6) * n^(3/4)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 06 2019

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Cf. A000010, A001157, A051953, A053650, A057660, A061255, A065764, A065827.

allocated for Ilya GutkovskiyExpansion of Product_{k>=1} 1/(1 - x^k)^(k-phi(k)), where phi() is the Euler totient function (A000010).

1, 0, 1, 1, 3, 2, 8, 5, 16, 15, 34, 30, 75, 66, 144, 150, 285, 292, 566, 585, 1062, 1170, 1988, 2205, 3729, 4159, 6755, 7785, 12214, 14147, 21957, 25560, 38709, 45839, 67884, 80747, 118332, 141244, 203614, 245330, 348396, 420971, 592439, 717659, 998248, 1215439, 1672544, 2040210, 2786687

0,5

Euler transform of A051953.

G.f.: exp(Sum_{k>=1} (sigma_2(k) - sigma_2(k^2)/sigma_1(k^2)) * x^k/k).

G.f.: exp(Sum_{k>=1} ( Sum_{d|k} cototient(d^2) ) * x^k/k).

nmax = 48; CoefficientList[Series[Product[1/(1 - x^k)^(k - EulerPhi[k]), {k, 1, nmax}], {x, 0, nmax}], x]

nmax = 48; CoefficientList[Series[Exp[Sum[(DivisorSigma[2, k] - DivisorSigma[2, k^2]/DivisorSigma[1, k^2]) x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x]

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^2 - EulerPhi[d^2], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 48}]

Cf. A000010, A001157, A051953, A053650, A057660, A061255.

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nonn

Ilya Gutkovskiy, Apr 22 2019

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allocated for Ilya Gutkovskiy

recycled

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