A344677 Number of partitions of n containing a prime number of primes and an arbitrary number of nonprimes.

0, 0, 0, 0, 1, 2, 4, 6, 9, 13, 20, 26, 36, 49, 68, 90, 120, 154, 201, 258, 330, 418, 532, 666, 834, 1041, 1290, 1592, 1958, 2404, 2935, 3588, 4345, 5278, 6366, 7692, 9215, 11096, 13230, 15853, 18831, 22477, 26580, 31620, 37247, 44145, 51851, 61247, 71681, 84445

Offset

0,6

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Example

a(6) = 4 because there are 4 partitions of 6 that contain a prime number of primes (including repetitions). These partitions are [3,3], [3,2,1], [2,2,2], [2,2,1,1].

Mathematica

nterms=50; Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]], 1, 0] &, IntegerPartitions[n]]], {n, 0, nterms-1}]
(* Second program: *)
seq[n_] := Module[{p}, p = 1/Product[1 - If[PrimeQ[k], y*x^k, 0] + O[x]^n, {k, 2, n}]; CoefficientList[Sum[If[PrimeQ[k], Coefficient[p, y, k], 0], {k, 2, n}]/QPochhammer[x + O[x]^n]/(p /. y -> 1), x]];
seq[50] (* Jean-François Alcover, May 27 2021, after Andrew Howroyd *)

Prog

(PARI) seq(n)={my(p=1/prod(k=2, n, 1 - if(isprime(k), y*x^k) + O(x*x^n))); Vec(sum(k=2, n, if(isprime(k), polcoef(p, k, y)))/eta(x+O(x*x^n))/subst(p, y, 1), -(n+1))} \\ Andrew Howroyd, May 26 2021

Crossrefs

Cf. A000040, A058698, A085755, A235945, A343753.

Sequence in context: A081659 A143586 A363457 * A241546 A370320 A280918

Adjacent sequences: A344674 A344675 A344676 * A344678 A344679 A344680

Keyword

nonn

Author

Paolo Xausa, May 26 2021

Status

approved