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A328556 Expansion of Product_{p prime, k>=1} (1 - x^(p^k)). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,17

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

FORMULA

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

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}]

CROSSREFS

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

Sequence in context: A170967 A035227 A049340 * A321888 A321750 A056929

Adjacent sequences:  A328553 A328554 A328555 * A328557 A328558 A328559

KEYWORD

sign,look

AUTHOR

Ilya Gutkovskiy, Nov 01 2019

STATUS

approved

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Last modified May 30 10:08 EDT 2020. Contains 334724 sequences. (Running on oeis4.)