A300672 Expansion of 1/(1 - Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k))).

1, 0, 1, 1, 3, 3, 8, 10, 23, 32, 64, 98, 187, 296, 543, 891, 1595, 2660, 4694, 7924, 13854, 23556, 40940, 69939, 121122, 207490, 358517, 615292, 1061635, 1824013, 3144404, 5406257, 9314645, 16021922, 27595176, 47478950, 81757104, 140691461, 242232918, 416890645, 717712748, 1235289624

Offset: 0

Author

Ilya Gutkovskiy, Mar 11 2018

Keywords

• nonn

Comments

Invert transform of A001222.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A001222, A022559, A112967, A129921, A180305, A293549, A300671.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*numtheory[bigomega](i), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Feb 11 2021

Mathematica

nmax = 41; CoefficientList[Series[1/(1 - Sum[Boole[PrimePowerQ[k]] x^k/(1 - x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 41; CoefficientList[Series[1/(1 - Sum[PrimeOmega[k] x^k, {k, 2, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PrimeOmega[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 41}]

Formula

G.f.: 1/(1 - Sum_{k>=2} A001222(k)*x^k).