A355158 -id:A355158 - cp's OEIS frontend

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

Omar E. Pol, Jun 28 2022

nonn

			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.

Index entries for sequences related to partitions

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

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

a(n) = my(nb=0); forpart(p=n, if (#select(x->!isprime(x), Vec(p)) != #p/2, nb++)); nb; \\ Michel Marcus, Jun 30 2022

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

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

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

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

Original entry on oeis.org

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

Omar E. Pol, Jun 24 2022

nonn

			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.

Index entries for sequences related to partitions

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

a(n) = my(nb=0); forpart(p=n, if (#select(isprime, Vec(p)) > #p/2, nb++)); nb; \\ Michel Marcus, Jun 25 2022

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

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

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

More terms from Alois P. Heinz, Jun 24 2022

cp's OEIS Frontend

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

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