A155515 Number of partitions of n into as many primes as nonprimes.

1, 0, 0, 1, 1, 0, 3, 2, 3, 5, 4, 8, 12, 10, 15, 23, 22, 33, 42, 47, 64, 79, 90, 122, 147, 169, 219, 264, 312, 387, 465, 546, 679, 799, 950, 1151, 1365, 1599, 1937, 2270, 2678, 3181, 3735, 4374, 5192, 6046, 7082, 8318, 9684, 11281, 13208, 15313, 17798, 20702, 23951

Offset: 0

Author

Reinhard Zumkeller, Jan 23 2009

Keywords

• nonn

Examples

			a(9) = #{6+3, 5+4, 5+2+1+1, 4+2+2+1, 2+2+2+1+1+1} = 5;
a(10) = #{8+2, 5+3+1+1, 4+3+2+1, 3+2+2+1+1+1} = 4.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for sequences related to partitions

Crossrefs

Cf. A000041, A000607, A002095, A027187, A355306.

Programs

Maple

b:= proc(n, i, t) local m; m:= n- `if`(t>0, t, -2*t); if m<0 then 0 elif n=0 then 1 elif i<3 then `if`(irem(m,3)=0, 1, 0) else b(n, i, t):= b(n-i, i, t+ `if`(isprime(i), 1, -1)) +b(n, i-1, t) fi end: a:= n-> b(n, n, 0): seq(a(n), n=0..60);  # Alois P. Heinz, Apr 30 2009

Mathematica

pnpQ[n_]:=Count[n,?PrimeQ]==Length[n]/2; Table[Count[ IntegerPartitions[ n], ?pnpQ],{n,60}] (* Harvey P. Dale, Feb 02 2014 *)
b[n_, i_, t_] := b[n, i, t] = Module[{m}, m = n - If[t > 0, t, -2t]; Which[m < 0, 0, n == 0, 1, i < 3, If[Mod[m, 3] == 0, 1, 0], True, b[n, i, t] = b[n-i, i, t + If[PrimeQ[i], 1, -1]] + b[n, i-1, t]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, May 30 2021, after Alois P. Heinz *)

PARI

parts(n)={1/(prod(k=1, n, 1 - if(isprime(k), y, 1/y)*x^k + O(x*x^n)))}
{my(n=60); apply(p->polcoeff(p,0), Vec(parts(n)))} \\ Andrew Howroyd, Dec 29 2017

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(55)]) # Michael S. Branicky, Jun 30 2022

Formula

a(n) = A000041(n) - A355306(n). - Omar E. Pol, Jun 30 2022