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

0,3

Links

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

Index entries for sequences related to partitions

Formula

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

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

Example

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.

Mathematica

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

Prog

(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
(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

Crossrefs

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

Sequence in context: A105669 A338917 A256963 * A019657 A134791 A095760

Adjacent sequences: A355303 A355304 A355305 * A355307 A355308 A355309

Keyword

nonn

Author

Omar E. Pol, Jun 28 2022

Status

approved