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%I #11 Jul 16 2015 22:19:41
%S 0,0,2,2,2,0,4,4,2,4,6,4,4,6,8,6,12,6,4,16,10,8,14,12,8,12,16,10,18,
%T 16,8,24,14,10,28,16,14,22,20,12,26,24,12,26,28,10,30,28,18,36,24,18,
%U 32,30,22,32,28,18,34,36,10,44,38,18,48,32,26,40,42,32,38,36,22,44
%N Number of decompositions of 2n into ordered sums of one prime and one nonprime.
%C A002372(n) + a(n) + A215462(n) = n.
%C Note: a(n) always even.
%C Conjecture: a(n) is never zero for n > 5, verified to 10^9.
%C Goldbach conjecture: a(n) + A215462(n) < n for all n > 2.
%F a(n) = convolve(p,c) + convolve(c,p) = 2*convolve(p,c) where p(n) = 1 if 2n+1 is prime and 0 otherwise, and c(n) = 1 if 2n+1 is nonprime and 0 otherwise.
%e n=15, 2*n=30, 2*n = { 3+27, 5+25, 29+1; 1+29, 25+5, 27+3 }, a(15) = 6
%e n=18, 2*n=36, 2*n = { 3+33, 11+25; 11+25, 33+3 }, a(18) = 4
%Y Cf. A002372, A215462.
%K nonn
%O 0,3
%A _Peter A. Lawrence_, Aug 11 2012