A054685 -id:A054685 - cp's OEIS frontend

A280596 Expansion of Product_{p prime, k>=2} (1 + x^(p^k)).

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 1, 1, 2, 0, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 3, 1, 1, 3, 3, 1, 1, 3, 3, 2, 1, 3, 4, 2, 1, 3, 4, 2, 1, 3, 4, 2, 1, 3, 4, 3, 1, 4, 4, 3, 1, 4, 5, 3, 2, 4, 6, 3, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 6, 4, 2, 4, 7, 4, 2, 4, 7, 5, 2

Offset: 0

Ilya Gutkovskiy, Jan 06 2017

nonn

Number of partitions of n into distinct proper prime powers (A246547).

			a(25) = 2 because we have [25] and [16, 9].

Eric Weisstein's World of Mathematics, Prime Power
Index entries for related partition-counting sequences

Cf. A054685, A106244, A246547, A280586.

nmax = 107; CoefficientList[Series[Product[(1 + Sign[PrimeOmega[k] - 1] Floor[1/PrimeNu[k]] x^k), {k, 2, nmax}], {x, 0, nmax}], x]

G.f.: Product_{p prime, k>=2} (1 + x^(p^k)).

A329097 Expansion of Product_{p prime, k>=1} 1 / (1 + x^(p^k)).

Original entry on oeis.org

1, 0, -1, -1, 0, 0, 1, 0, 0, -1, 1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -2, 3, -4, 3, -4, 5, -5, 6, -6, 7, -8, 9, -9, 11, -12, 13, -16, 15, -17, 20, -22, 23, -26, 29, -30, 35, -38, 40, -45, 50, -52, 58, -65, 69, -75, 82, -89, 96, -107, 114, -123, 135, -145, 158, -170, 185, -200, 216, -232, 251

Offset: 0

Ilya Gutkovskiy, Nov 04 2019

sign

Convolution inverse of A054685.

Cf. A023894, A048165, A054685, A246655, A328556.

nmax = 70; CoefficientList[Series[Product[1/(1 + Boole[PrimePowerQ[k]] x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d) Boole[PrimePowerQ[d]] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 70}]

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

cp's OEIS Frontend

A280596 Expansion of Product_{p prime, k>=2} (1 + x^(p^k)).

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