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

1,6

Comments

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

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Subsequence of A224708.

Cf. A010051.

Sequence in context: A284511 A262976 A349608 * A309965 A366385 A070172

Adjacent sequences: A224706 A224707 A224708 * A224710 A224711 A224712

Keyword

nonn

Author

J. Stauduhar, Apr 16 2013

Status

approved