A299730 -id:A299730 - cp's OEIS frontend

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

J. Stauduhar, Feb 18 2018

nonn

			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.

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

Cf. A222656, A299730, A299732.

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 *)

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.
```

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

A299732 a(n) has exactly (a(n) - n) / 2 partitions with exactly (a(n) - n) / 2 prime parts.

Original entry on oeis.org

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

Offset: 0

J. Stauduhar, Feb 18 2018

nonn

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?

			For n = 3: A299731(3) = 5. a(3) = 2*5 + 3 = 13. The five partitions of 13 that have exactly five prime parts are: (5,2,2,2,2), (3,3,3,2,2), (3,3,2,2,2,1), (3,2,2,2,2,1,1), and (2,2,2,2,2,1,1,1), so a(3) = 13.

J. Stauduhar, Python program.

Cf. A222656, A299730, A299731.

Python
```
# See Stauduhar link.
```

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

cp's OEIS Frontend

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

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