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A328556
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Expansion of Product_{p prime, k>=1} (1 - x^(p^k)).
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2
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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
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OFFSET
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0,17
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COMMENTS
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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.
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LINKS
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FORMULA
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G.f.: Product_{k>=1} (1 - x^A246655(k)).
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MAPLE
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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:
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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