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

0,7

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

Crossrefs

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

Sequence in context: A071166 A140358 A130727 * A023605 A223542 A182003

Adjacent sequences: A155512 A155513 A155514 * A155516 A155517 A155518

Keyword

nonn

Author

Reinhard Zumkeller, Jan 23 2009

Status

approved