A281545 Expansion of Sum_{k>=2} x^prime(k)/(1 + x^prime(k)) * Product_{k>=2} (1 + x^prime(k)).

0, 0, 1, 0, 1, 0, 1, 2, 0, 2, 1, 2, 1, 2, 3, 4, 1, 4, 4, 4, 6, 4, 7, 6, 6, 8, 9, 8, 10, 6, 13, 12, 12, 14, 15, 16, 16, 18, 23, 22, 19, 24, 24, 30, 28, 30, 33, 34, 34, 40, 44, 46, 44, 46, 58, 56, 60, 64, 65, 68, 70, 80, 86, 88, 87, 94, 101, 112, 114, 116, 125, 130, 132, 148, 159, 162, 163, 168, 190, 196

Offset: 1

Author

Ilya Gutkovskiy, Jan 23 2017

Keywords

• nonn

Comments

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

Examples

			a(23) = 7 because we have [23], [13, 7, 3], [11, 7, 5] and 1 + 3 + 3 = 7.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..1000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A024938, A024939, A065091.

Programs

Mathematica

nmax = 80; 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]]

PARI

sumparts(n, pred)={sum(k=1, n, 1 - 1/(1+pred(k)*x^k) + 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

Formula

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