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

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

0,1

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.

Example

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.

Prog

(Python) # See Stauduhar link.

Crossrefs

Cf. A222656, A299730, A299731.

Sequence in context: A054254 A025216 A076059 * A169952 A025279 A169954

Adjacent sequences: A299729 A299730 A299731 * A299733 A299734 A299735

Keyword

nonn

Author

J. Stauduhar, Feb 18 2018

Status

approved