%I #30 Mar 24 2018 06:08:37

%S 1,2,3,5,8,12,18,25,35,50,69,93,126,167,220,290,377,486,627,800,1017,

%T 1290,1623,2032,2542,3161,3917,4843,5960,7312,8957,10925,13291,16139,

%U 19534,23588,28437,34180,41000,49099,58657,69941,83269,98917,117314,138930

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

%H Alois P. Heinz, <a href="/A299731/b299731.txt">Table of n, a(n) for n = 0..2000</a>

%H J. Kelleher, B. O'Sullivan, <a href="https://arxiv.org/abs/0909.2331">Generating All Partitions: A Comparison Of Two Encodings</a>, arXiv:0909.2331 [cs.DS], 2009, 2014.

%H J. Stauduhar, <a href="/A299731/a299731.py.txt">Python program</a>

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

%e 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.

%t zip[f_, x_, y_, z_] := With[{m = Max[Length[x], Length[y]]}, Thread[f[ PadRight[x, m, z], PadRight[y, m, z]]]];

%t 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]];

%t a[n_] := b[3 n, 3 n][[n + 1]];

%t Table[a[n], {n, 0, 45}] (* _Jean-François Alcover_, Mar 16 2018, after _Alois P. Heinz_ *)

%o (Python) See Stauduhar link.

%o (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

%Y Cf. A222656, A299730, A299732.

%K nonn

%O 0,2

%A _J. Stauduhar_, Feb 18 2018