A116449 Number of partitions of n into an equal number of prime and composite parts.

1, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 4, 3, 4, 4, 6, 6, 11, 11, 13, 17, 20, 26, 32, 37, 43, 55, 63, 81, 92, 111, 126, 159, 183, 225, 259, 307, 357, 430, 497, 589, 683, 797, 929, 1093, 1270, 1478, 1712, 1979, 2303, 2665, 3086, 3556, 4102, 4716, 5444, 6256, 7194, 8243, 9456, 10824

Offset: 0

Author

Reinhard Zumkeller, Feb 16 2006

Keywords

• nonn

Comments

a(n) = A002865(n) - A116450(n).

Examples

			a(14) = #{2+2*2*3, (2+2)+(2*3+2*2), 5+3*3, (3+3)+(2*2+2*2)} =
4;
a(15) = #{3+2*2*3, 5+2*5, (2+3)+(2*2+2*3), 7+2*2*2,
(2+5)+(2*2+2*2), 11+2*2} = 6.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Programs

Mathematica

terms = 61;
pp = 1/(Product[1 - If[PrimeQ[k], y, 1/y]*x^k, {k, 2, terms-3}] + O[x]^terms) // Normal;
Take[Expand[pp ], terms-5] // CoefficientList[#, x]& (* Jean-François Alcover, Dec 30 2017, after Andrew Howroyd *)

PARI

parts(n)={1/(prod(k=2, 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

Extensions

a(0)=1 from Andrew Howroyd, Dec 29 2017