A224709 The number of unordered partitions {a,b} of the even numbers 2n such that a and b are composite.

0, 0, 0, 1, 1, 2, 2, 3, 4, 4, 4, 6, 5, 6, 8, 7, 8, 10, 8, 10, 12, 11, 11, 14, 13, 13, 16, 14, 15, 19, 15, 18, 20, 17, 20, 22, 20, 21, 24, 22, 22, 27, 23, 24, 30, 25, 26, 30, 27, 30, 33, 30, 30, 34, 32, 33, 37, 33, 33, 41, 33, 36, 42, 36, 40, 43, 39, 40, 44

Offset: 1

Author

J. Stauduhar, Apr 16 2013

Keywords

• nonn

Comments

Conjecture: a(3n+9) > a(3n+8) and a(3n+10) < a(3n+9) for n>=1. - Anthony Browne, Jun 26 2016

Examples

			For n=6, 2*6=12 and the partitions of 12 are (1,11),(2,10),(3,9),(4,8),(5,7),(6,6). Of these, 2 are composite pairs, namely (4,8),(6,6) so a(6)=2.
		

Links

• J. Stauduhar, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A224708.

Cf. A010051.

Programs

Mathematica

Table[Count[Transpose@ {#, 2 n - #} &@ Range@ n, w_ /; Times @@ Boole@ Map[CompositeQ, w] > 0], {n, 69}] (* Michael De Vlieger, Jun 26 2016 *)

PARI

a(n) = sum(k=1, n-1, (1-isprime(k+1))*(1-isprime(2*n-k-1))); \\ Michel Marcus, Apr 11 2022

Formula

a(n) = Sum_{k=1..n-1} (1-A010051(k+1))(1-A010051(2n-k-1)). - Anthony Browne, Jun 26 2016