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%I #16 Jul 21 2021 19:18:13
%S 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,
%T 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,
%U 12,16,8,18,6,18,14,17,9,19,7,18,11,19,11,21
%N The number of unordered partitions {a, b} of n such that a or b is composite and the other is prime.
%H J. Stauduhar, <a href="/A224712/b224712.txt">Table of n, a(n) for n = 1..10000</a>
%e 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.
%e For n = 9, we have partitions {6, 3} and {5, 4}, so a(9) = 2.
%t Table[Length[Select[Range[2, Floor[n/2]], (PrimeQ[#] && Not[PrimeQ[n - #]]) || (Not[PrimeQ[#]] && PrimeQ[n - #]) &]], {n, 80}] (* _Alonso del Arte_, Apr 21 2013 *)
%t Table[Count[IntegerPartitions[n,{2}],_?(FreeQ[#,1]&&Total[Boole[ PrimeQ[ #]]] == 1&)],{n,80}] (* _Harvey P. Dale_, Jul 21 2021 *)
%o (PARI) a(n)=my(s);forprime(p=2,n-4,s+=!isprime(n-p));s \\ _Charles R Greathouse IV_, Apr 30 2013
%Y Cf. A062602 (allows 1 as well as composites), A224708 (a and b are both composite).
%K nonn,easy
%O 1,9
%A _J. Stauduhar_, Apr 20 2013
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