A279412 Expansion of Sum_{k>=1} prime(k)*x^prime(k)/(1 + x^prime(k)) * Product_{k>=1} (1 + x^prime(k)).
0, 2, 3, 0, 10, 0, 14, 8, 9, 20, 11, 24, 26, 28, 30, 48, 34, 72, 57, 80, 84, 88, 115, 120, 125, 156, 135, 168, 203, 180, 279, 224, 297, 306, 315, 396, 407, 418, 507, 480, 574, 630, 645, 748, 720, 828, 893, 960, 1029, 1150, 1122, 1300, 1378, 1458, 1650, 1624, 1824, 1856, 2065, 2220, 2379, 2480, 2646, 2816, 2925
Offset: 1
Keywords
Examples
a(12) = 24 because we have [7, 5], [7, 3, 2] and 2*12 = 24.
Links
- Eric Weisstein's World of Mathematics, Prime Partition
- Index entries for related partition-counting sequences
Programs
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Mathematica
nmax = 65; Rest[CoefficientList[Series[Sum[Prime[k] x^Prime[k]/(1 + x^Prime[k]), {k, 1, nmax}] Product[1 + x^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x]] nmax = 65; Rest[CoefficientList[Series[x D[Product[1 + x^Prime[k], {k, 1, nmax}], x], {x, 0, nmax}], x]]
Formula
G.f.: Sum_{k>=1} prime(k)*x^prime(k)/(1 + x^prime(k)) * Product_{k>=1} (1 + x^prime(k)).
G.f.: x*f'(x), where f(x) = Product_{k>=1} (1 + x^prime(k)).
a(n) = n*A000586(n).
Comments