A329097 Expansion of Product_{p prime, k>=1} 1 / (1 + x^(p^k)).

1, 0, -1, -1, 0, 0, 1, 0, 0, -1, 1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -2, 3, -4, 3, -4, 5, -5, 6, -6, 7, -8, 9, -9, 11, -12, 13, -16, 15, -17, 20, -22, 23, -26, 29, -30, 35, -38, 40, -45, 50, -52, 58, -65, 69, -75, 82, -89, 96, -107, 114, -123, 135, -145, 158, -170, 185, -200, 216, -232, 251

Offset: 0

Author

Ilya Gutkovskiy, Nov 04 2019

Keywords

• sign

Comments

Convolution inverse of A054685.

Crossrefs

Cf. A023894, A048165, A054685, A246655, A328556.

Programs

Mathematica

nmax = 70; CoefficientList[Series[Product[1/(1 + Boole[PrimePowerQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) Boole[PrimePowerQ[d]] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 70}]

Formula

G.f.: Product_{k>=1} 1 / (1 + x^A246655(k)).