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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,7

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(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