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

0,6

Links

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

Index entries for sequences related to partitions

Formula

a(n) = A000041(n) - A155515(n) - A355158(n).

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

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A211516 A323430 A291941 * A086341 A176513 A128424

Adjacent sequences: A355222 A355223 A355224 * A355226 A355227 A355228

Keyword

nonn

Author

Omar E. Pol, Jun 24 2022

Extensions

More terms from Alois P. Heinz, Jun 24 2022

Status

approved