A299731 Number of partitions of 3*n that have exactly n prime parts.

1, 2, 3, 5, 8, 12, 18, 25, 35, 50, 69, 93, 126, 167, 220, 290, 377, 486, 627, 800, 1017, 1290, 1623, 2032, 2542, 3161, 3917, 4843, 5960, 7312, 8957, 10925, 13291, 16139, 19534, 23588, 28437, 34180, 41000, 49099, 58657, 69941, 83269, 98917, 117314, 138930

Offset: 0

Author

J. Stauduhar, Feb 18 2018

Keywords

• nonn

Examples

			For n = 3: the five partitions of 3 * 3 = 9 that have exactly three prime parts are (5, 2, 2), (3, 3, 3), (3, 3, 2, 1), (3, 2, 2, 1, 1), and (2, 2, 2, 1, 1, 1), so a(3) = 5.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000
• J. Kelleher, B. O'Sullivan, Generating All Partitions: A Comparison Of Two Encodings, arXiv:0909.2331 [cs.DS], 2009, 2014.
• J. Stauduhar, Python program

Crossrefs

Cf. A222656, A299730, A299732.

Programs

Mathematica

zip[f_, x_, y_, z_] := With[{m = Max[Length[x], Length[y]]}, Thread[f[ PadRight[x, m, z], PadRight[y, m, z]]]];
b[n_, i_] := b[n, i] = Module[{j, pc}, Which[n == 0, {1}, i < 1, {0}, True, pc = {}; For[j = 0, j <= n/i, j++, pc = zip[Plus, pc, Join[If[PrimeQ[i], Array[0 &, j], {}], b[n - i*j, i - 1]], 0]]; pc]];
a[n_] := b[3 n, 3 n][[n + 1]];
Table[a[n], {n, 0, 45}] (* Jean-François Alcover, Mar 16 2018, after Alois P. Heinz *)

PARI

a(n) = {my(nb = 0); forpart(p=3*n, if (sum(k=1, #p, isprime(p[k])) == n, nb++);); nb;} \\ Michel Marcus, Mar 22 2018

Python

See Stauduhar link.

Formula

a(n) = A222656(3*n,n).