A338826 G.f.: (1/(1 + x)) * Product_{k>=1} 1/(1 + x^prime(k)).

1, -1, 0, -1, 2, -2, 2, -3, 4, -4, 5, -7, 8, -9, 11, -13, 15, -18, 21, -24, 28, -32, 37, -43, 49, -55, 63, -72, 81, -92, 104, -117, 131, -147, 166, -185, 206, -231, 257, -285, 317, -353, 391, -432, 478, -528, 583, -643, 708, -778, 855, -940, 1031, -1130, 1238, -1354

Offset: 0

Author

Ilya Gutkovskiy, Dec 02 2020

Keywords

• sign

Comments

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

Convolution inverse of A036497.

Crossrefs

Cf. A000586, A000607, A008578, A036497, A046675, A048165, A298602.

Programs

Mathematica

nmax = 55; CoefficientList[Series[(1/(1 + x)) Product[1/(1 + x^Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[k, (-1)^(k/#) # &, PrimeQ[#] || # == 1 &] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 55}]

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * A048165(k).