A355306 Number of partitions of n in which the number of prime parts is not equal to the number of nonprime parts.

0, 1, 2, 2, 4, 7, 8, 13, 19, 25, 38, 48, 65, 91, 120, 153, 209, 264, 343, 443, 563, 713, 912, 1133, 1428, 1789, 2217, 2746, 3406, 4178, 5139, 6296, 7670, 9344, 11360, 13732, 16612, 20038, 24078, 28915, 34660, 41402, 49439, 58887, 69983, 83088, 98476, 116436, 137589, 162244, 191018

Offset: 0

Author

Omar E. Pol, Jun 28 2022

Keywords

• nonn

Examples

			For n = 6 the partitions of 6 in which the number of prime parts is not equal to the number of nonprime parts are [6], [3, 3], [2, 2, 2], [3, 2, 1], [4, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1], there are eight of these partitions so a(6) = 8.
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A000040, A000041, A000607, A002095, A002096, A018252, A155515, A355158, A355225.

Programs

Mathematica

Array[Count[IntegerPartitions[#], ?(#1 - #2 != #2 & @@ {Length[#], Count[#, ?PrimeQ]} &)] &, 51, 0] (* Michael De Vlieger, Jul 15 2022 *)

PARI

a(n) = my(nb=0); forpart(p=n, if (#select(x->!isprime(x), Vec(p)) != #p/2, nb++)); nb; \\ Michel Marcus, Jun 30 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).

a(n) = A355158(n) + A355225(n).