A299730 -id:A299730

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A299731

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

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.

KEYWORD

nonn

AUTHOR

J. Stauduhar, Feb 18 2018

STATUS

approved

A299732

a(n) has exactly (a(n) - n) / 2 partitions with exactly (a(n) - n) / 2 prime parts.

+10
2

2, 5, 8, 13, 20, 29, 42, 57, 78, 109, 148, 197, 264, 347, 454, 595, 770, 989, 1272, 1619, 2054, 2601, 3268, 4087, 5108, 6347, 7860, 9713, 11948, 14653, 17944, 21881, 26614, 32311, 39102, 47211, 56910, 68397, 82038, 98237, 117354, 139923, 166580, 197877, 234672

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

If B={b(n)} is the complement of A299731 then no number exists that has exactly b(n) partitions that have exactly b(n) prime parts, so this sequence lists only those numbers that can have the equality property.

Up to a(44) = 234672 (currently, the last term), except for 2,5,8, and 29, every term is the sum of distinct previous terms. Will this be true for all new terms?

LINKS

Table of n, a(n) for n=0..44.

J. Stauduhar, Python program.

FORMULA

a(n) = 2*A299731(n) + n = 2*A222656(3*n,n) + n.