A224708 The number of unordered partitions {a,b} of n such that a and b are composite.

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 3, 1, 4, 2, 4, 2, 4, 2, 6, 3, 5, 3, 6, 4, 8, 5, 7, 5, 8, 5, 10, 6, 8, 7, 10, 7, 12, 8, 11, 8, 11, 8, 14, 9, 13, 9, 13, 10, 16, 11, 14, 11, 15, 12, 19, 13, 15, 13, 18, 13, 20, 14, 17, 15, 20, 15, 22, 16, 20, 16, 21

Offset: 1

Author

J. Stauduhar, Apr 16 2013

Keywords

• nonn
• easy

Comments

For n > 11, a(n) > 0. - Geoffrey Critzer, Jan 31 2015

Last occurrence of n is a(A014092(n+4)). - Anthony Browne, May 25 2016

Examples

			For n=8, in the set {{7,1},{6,2},{5,3},{4,4}}, {4,4} is the only partition {a,b} where a and b are both composite, so a(8)=1.
For n=12, we have partitions {8,4} and {6,6}, so a(12)=2.
		

Links

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

Crossrefs

Cf. A010051, A014092, A045917, A066247.

Programs

Mathematica

nn = 76; Rest[Transpose[CoefficientList[Series[Product[1/(1 - y x^i), {i, Select[Range[2, nn], ! PrimeQ[#] &]}], {x,0,nn}], {x, y}]][[3]]] (* Geoffrey Critzer, Jan 31 2015 *)
f[n_] := Count[ PrimeQ@ Rest@ IntegerPartitions[ n, {2}], {False, False}]; Array[f, 76] (* Robert G. Wilson v, Feb 04 2015 *)

Formula

a(2*n) - a(2*n+1) + A010051(n) = A045917(n). - Anthony Browne, May 03 2016

a(A014092(n+4)) = n. - Anthony Browne, May 25 2016