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

%I A328556 #25 Nov 03 2019 12:21:02
%S A328556 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,
%T A328556 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,
%U A328556 -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
%N A328556 Expansion of Product_{p prime, k>=1} (1 - x^(p^k)).
%C A328556 Convolution inverse of A023894.
%C A328556 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).
%C A328556 Conjecture: the last zero (38th) occurs at n = 340.
%H A328556 Robert Israel, <a href="/A328556/b328556.txt">Table of n, a(n) for n = 0..10000</a>
%F A328556 G.f.: Product_{k>=1} (1 - x^A246655(k)).
%p A328556 N:= 100: # for a(0)..a(N)
%p A328556 R:= 1:
%p A328556 p:= 1:
%p A328556 do
%p A328556   p:= nextprime(p);
%p A328556   if p > N then break fi;
%p A328556   for k from 1 to floor(log[p](N)) do
%p A328556     R:= series(R*(1-x^(p^k)),x,N+1)
%p A328556   od;
%p A328556 od:
%p A328556 seq(coeff(R,x,j),j=0..N); # _Robert Israel_, Nov 03 2019
%t A328556 nmax = 90; CoefficientList[Series[Product[(1 - Boole[PrimePowerQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%t A328556 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}]
%Y A328556 Cf. A023894, A046675, A054685, A246655, A292561.
%K A328556 sign,look
%O A328556 0,17
%A A328556 _Ilya Gutkovskiy_, Nov 01 2019