A328556 Expansion of Product_{p prime, k>=1} (1 - x^(p^k)).

1, 0, -1, -1, -1, 0, 1, 1, 0, 0, 1, 1, 1, 0, -1, -1, -2, -1, 0, 0, 1, 1, 0, -1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, -3, -3, -1, 1, 1, 0, -1, -1, 2, 2, 0, 1, -1, 0, 1, 0, -1, 0, 1, 0, 0, -2, -3, -1, -1, 0, 2, 0, 1, 3, 0, 1, 3, 1, -3, -2, -3, -2, 3, 2, -1, 0, -2, 1, 1, -2, -1, 1, 2, 2, 3, -1, -2, 4

Offset: 0

Author

Ilya Gutkovskiy, Nov 01 2019

Keywords

• sign
• look

Comments

Convolution inverse of A023894.

The difference between the number of partitions of n into an even number of distinct prime power parts and the number of partitions of n into an odd number of distinct prime power parts (1 excluded).

Conjecture: the last zero (38th) occurs at n = 340.

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A023894, A046675, A054685, A246655, A292561.

Programs

Maple

N:= 100: # for a(0)..a(N)
R:= 1:
p:= 1:
do
  p:= nextprime(p);
  if p > N then break fi;
  for k from 1 to floor(log[p](N)) do
    R:= series(R*(1-x^(p^k)),x,N+1)
  od;
od:
seq(coeff(R,x,j),j=0..N); # Robert Israel, Nov 03 2019

Mathematica

nmax = 90; CoefficientList[Series[Product[(1 - Boole[PrimePowerQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, -Sum[Sum[Boole[PrimePowerQ[d]] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 90}]

Formula

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