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

0,5

Links

Table of n, a(n) for n=0..50.

Index entries for sequences related to partitions

Formula

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

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

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A030403 A361663 A369575 * A034403 A215082 A320690

Adjacent sequences: A355155 A355156 A355157 * A355159 A355160 A355161

Keyword

nonn

Author

Omar E. Pol, Jun 24 2022

Extensions

More terms from Michel Marcus, Jun 25 2022

Status

approved