A355225 Number of partitions of n that contain more prime parts than nonprime parts.

0, 0, 1, 1, 1, 3, 3, 5, 7, 9, 14, 19, 23, 34, 46, 56, 77, 99, 126, 164, 208, 260, 336, 416, 520, 654, 809, 995, 1237, 1514, 1856, 2274, 2761, 3354, 4078, 4918, 5931, 7153, 8572, 10272, 12298, 14663, 17469, 20787, 24643, 29210, 34568, 40797, 48113, 56664, 66573

Offset: 0

Author

Omar E. Pol, Jun 24 2022

Keywords

• nonn

Examples

			For n = 8 the partitions of 8 that contain more prime parts than nonprime parts are [5, 3], [3, 3, 2], [4, 2, 2], [2, 2, 2, 2], [5, 2, 1], [3, 2, 2, 1], [2, 2, 2, 1, 1]. There are seven of these partitions so a(8) = 7.
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A000040, A000041, A000607, A002095, A002096, A018252, A155515, A235945, A355158, A355306.

Programs

PARI

a(n) = my(nb=0); forpart(p=n, if (#select(isprime, 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 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) - A355158(n).

a(n) = A355306(n) - A355158(n).

Extensions

More terms from Alois P. Heinz, Jun 24 2022