A355158 Number of partitions of n that contain more nonprime parts than prime parts.

0, 1, 1, 1, 3, 4, 5, 8, 12, 16, 24, 29, 42, 57, 74, 97, 132, 165, 217, 279, 355, 453, 576, 717, 908, 1135, 1408, 1751, 2169, 2664, 3283, 4022, 4909, 5990, 7282, 8814, 10681, 12885, 15506, 18643, 22362, 26739, 31970, 38100, 45340, 53878, 63908, 75639, 89476, 105580, 124445

Offset: 0

Author

Omar E. Pol, Jun 24 2022

Keywords

• nonn

Examples

			For n = 8 the partitions of 8 that contain more nonprime parts than prime parts are [8], [4, 4], [4, 3, 1], [6, 1, 1], [4, 2, 1, 1], [5, 1, 1, 1], [3, 2, 1, 1, 1], [4, 1, 1, 1, 1], [2, 2, 1, 1, 1, 1], [3, 1, 1, 1, 1, 1], [2, 1, 1, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1, 1, 1]. There are 12 of these partitions so a(8) = 12.
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A000040, A000041, A000607, A002095, A002096, A018252, A155515, A355225, A355306.

Programs

PARI

a(n) = my(nb=0); forpart(p=n, if (#select(x->!isprime(x), Vec(p)) > #p/2, nb++)); nb; \\ Michel Marcus, Jun 25 2022

Python

from sympy import isprime
from sympy.utilities.iterables import partitions
def c(p): return 2*sum(p[i] for i in p if not isprime(i)) > sum(p.values())
def a(n): return sum(1 for p in partitions(n) if c(p))
print([a(n) for n in range(51)]) # Michael S. Branicky, Jun 28 2022

Formula

a(n) = A000041(n) - A155515(n) - A355225(n).

a(n) = A355306(n) - A355225(n).

Extensions

More terms from Michel Marcus, Jun 25 2022