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

0,10

Comments

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

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Sequence in context: A093779 A231898 A276638 * A316433 A071046 A324757

Adjacent sequences: A116446 A116447 A116448 * A116450 A116451 A116452

Keyword

nonn

Author

Reinhard Zumkeller, Feb 16 2006

Extensions

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

Status

approved