A224712 The number of unordered partitions {a, b} of n such that a or b is composite and the other is prime.

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

Offset: 1

Author

J. Stauduhar, Apr 20 2013

Keywords

• nonn
• easy

Examples

			For n = 6, in the set {{5, 1}, {4, 2}, {3, 3}}, {4, 2} is the only partition that satisfies the requirements, so a(6) = 1.
For n = 9, we have partitions {6, 3} and {5, 4}, so a(9) = 2.
		

Links

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

Crossrefs

Cf. A062602 (allows 1 as well as composites), A224708 (a and b are both composite).

Programs

Mathematica

Table[Length[Select[Range[2, Floor[n/2]], (PrimeQ[#] && Not[PrimeQ[n - #]]) || (Not[PrimeQ[#]] && PrimeQ[n - #]) &]], {n, 80}] (* Alonso del Arte, Apr 21 2013 *)
Table[Count[IntegerPartitions[n,{2}],?(FreeQ[#,1]&&Total[Boole[ PrimeQ[ #]]] == 1&)],{n,80}] (* _Harvey P. Dale, Jul 21 2021 *)

PARI

a(n)=my(s);forprime(p=2,n-4,s+=!isprime(n-p));s \\ Charles R Greathouse IV, Apr 30 2013