A281544 Expansion of Sum_{k>=2} x^prime(k)/(1 - x^prime(k)) / Product_{k>=2} (1 - x^prime(k)).

0, 0, 1, 0, 1, 2, 1, 2, 3, 4, 4, 6, 7, 8, 11, 12, 15, 18, 20, 26, 29, 34, 40, 46, 54, 62, 71, 82, 94, 106, 122, 138, 157, 178, 201, 226, 254, 286, 321, 360, 402, 448, 501, 558, 619, 690, 764, 846, 938, 1036, 1145, 1264, 1392, 1532, 1687, 1854, 2036, 2234, 2448, 2680, 2934, 3210, 3507, 3828, 4178, 4554, 4961, 5404

Offset

1,6

Comments

Total number of parts in all partitions of n into odd primes.

Convolution of A005087 and A099773.

Links

Table of n, a(n) for n=1..68.

Index entries for related partition-counting sequences

Formula

G.f.: Sum_{k>=2} x^prime(k)/(1 - x^prime(k)) / Product_{k>=2} (1 - x^prime(k)).

Example

a(14) = 8 because we have [11, 3], [7, 7], [5, 3, 3, 3] and 2 + 2 + 4 = 8.

Mathematica

nmax = 68; Rest[CoefficientList[Series[Sum[x^Prime[k]/(1 - x^Prime[k]), {k, 2, nmax}]/Product[1 - x^Prime[k], {k, 2, nmax}], {x, 0, nmax}], x]]

Prog

(PARI)
sumparts(n, pred)={sum(k=1, n, 1/(1-pred(k)*x^k) - 1 + O(x*x^n))/prod(k=1, n, 1-pred(k)*x^k + O(x*x^n))}
{my(n=60); Vec(sumparts(n, v->v>2 && isprime(v)), -n)} \\ Andrew Howroyd, Dec 28 2017

Crossrefs

Cf. A005087, A065091, A084993, A099773.

Sequence in context: A030562 A238219 A026833 * A056882 A035534 A241416

Adjacent sequences: A281541 A281542 A281543 * A281545 A281546 A281547

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 23 2017

Status

approved