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

0,2

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

Formula

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

Example

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.

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 *)

Prog

(Python) See Stauduhar link.
(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

Crossrefs

Cf. A222656, A299730, A299732.

Sequence in context: A127884 A105858 A372040 * A252864 A039899 A039901

Adjacent sequences: A299728 A299729 A299730 * A299732 A299733 A299734

Keyword

nonn

Author

J. Stauduhar, Feb 18 2018

Status

approved