A123030 Partial sums of A038538.

1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 20, 26, 27, 29, 30, 32, 33, 34, 35, 38, 40, 41, 44, 46, 47, 48, 49, 57, 58, 59, 60, 64, 65, 66, 67, 70, 71, 72, 73, 75, 77, 78, 79, 85, 87, 89, 90, 92, 93, 96, 97, 100, 101, 102, 103, 105, 106, 107, 109, 122, 123, 124, 125, 127, 128

Offset: 1

Author

Jonathan Vos Post, Jul 07 2008

Keywords

• nonn

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, pp. 274-276.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A038538, A244285 (A_1*B_1).

Programs

Mathematica

With[{emax = 7}, f[e_] := f[e] = Coefficient[Series[Product[1/(1 - x^(j*k^2)), {k, 1, Floor[Sqrt[emax]] + 1}, {j, 1, Floor[emax/k^2] + 1}], {x, 0, emax}], x, e]; a[1] = 1; a[n_] := Times @@ f /@ FactorInteger[n][[;; , 2]]; Accumulate@ Array[a, 2^emax]] (* Amiram Eldar, Jan 31 2024, using code by Vaclav Kotesovec at A004101 *)

Formula

a(n) = A_1*B_1*n + A_2*B_2*n^(1/2) + A_3*B_3*n^(1/3) + O(n^(50/199+eps)), where A_k = Product_{m>=1, m!=k} zeta(m/k) and B_k = Product_{r>=1, m>=2} zeta(r*m^2/k) (Finch, 2003). - Amiram Eldar, Jan 31 2024