n = 22 has floor(n/2) = 11 partitions of form n = a+b; 3 partitions are of prime+prime [3+19 = 5+17 = 11+11], 3 partitions are of prime+nonprime [2+20 = 7+15 = 13+9], 5 partitions are of nonprime+nonprime [1+21 = 4+18 = 6+16 = 8+14 = 10+12]. So a(22) = 5.

MATHEMATICA

Table[Count[Transpose@ {#, n - #} &@ Range[Floor[n/2]], w_ /; Times @@ Boole@ Map[! PrimeQ@ # &, w] == 1], {n, 83}] (* Michael De Vlieger, Jul 04 2016 *)