%I
%S 0,0,0,0,0,0,1,1,2,1,2,1,2,1,2,2,4,2,2,2,3,2,3,2,2,2,3,1,3,2,2,2,4,2,
%T 2,4,4,1,3,2,3,2,3,3,4,3,4,3,3,3,5,2,4,4,2,2,6,2,2,4,1,1,5,4,5,4,4,2,
%U 4,3,3,3,4,4,5,4,5,4,3,2,4,2,3,6,5,3,6,3,5,5,2,3,9,3,3,5,3,1,6,3
%N Number of ways to write n = (1 + (n mod 2))*p + q with p < n/2 such that p, q and prime(p)  p + 1 are all prime.
%C Conjecture: a(n) > 0 for all n > 6.
%C This implies both Goldbach's conjecture (A045917) and Lemoine's conjecture (A046927). For primes p with prime(p)  p + 1 also prime, see A234695.
%H ZhiWei Sun, <a href="/A235189/b235189.txt">Table of n, a(n) for n = 1..10000</a>
%H Z.W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e a(10) = 1 since 10 = 3 + 7 with 3, 7 and prime(3)  3 + 1 = 3 all prime.
%e a(28) = 1 since 28 = 5 + 23 with 5, 23 and prime(5)  4 = 7 all prime.
%e a(61) = 1 since 61 = 2*7 + 47 with 7, 47 and prime(7)  6 = 11 all prime.
%e a(98) = 1 since 98 = 31 + 67 with 31, 67 and prime(31)  30 = 97 all prime.
%t p[n_]:=PrimeQ[Prime[n]n+1]
%t a[n_]:=Sum[If[p[Prime[k]]&&PrimeQ[n(1+Mod[n,2])*Prime[k]],1,0],{k,1,PrimePi[(n1)/2]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A045917, A046927, A234694, A234695, A235187.
%K nonn
%O 1,9
%A _ZhiWei Sun_, Jan 04 2014
