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%I #6 Jan 15 2014 04:44:48
%S 0,0,0,1,2,2,3,3,3,3,4,2,3,2,2,4,3,3,3,3,5,5,4,3,4,4,3,5,3,1,5,5,3,5,
%T 2,4,4,3,5,4,4,4,6,5,4,6,5,3,6,6,6,5,2,3,4,3,5,5,4,5,6,4,3,6,4,3,6,4,
%U 4,5,3,5,3,5,6,6,5,3,6,4,2,4,1,4,5,4,5,7,5,4,6,9,5,6,4,2,6,6,2,6
%N Number of ordered ways to write n = p + q with p a term of A234695 and q a term of A235592.
%C Conjecture: a(n) > 0 for all n > 3, and a(n) = 1 only for n = 4, 30, 83.
%H Zhi-Wei Sun, <a href="/A235703/b235703.txt">Table of n, a(n) for n = 1..10000</a>
%e a(4) = 1 since 4 = 2 + 2 with 2, prime(2) - 2 + 1 = 2 and 2*3 - prime(2) = 3 all prime.
%e a(30) = 1 since 30 = 3 + 27 with 3, prime(3) - 3 + 1 = 3 and 27*28 - prime(27) = 756 - 103 = 653 all prime.
%e a(83) = 1 since 83 = 13 + 70 with 13, prime(13) - 13 + 1 = 29 and 70*71 - prime(70) = 4970 - 349 = 4621 all prime.
%t p[n_]:=PrimeQ[Prime[n]-n+1]
%t q[n_]:=PrimeQ[n(n+1)-Prime[n]]
%t a[n_]:=Sum[If[p[Prime[k]]&&q[n-Prime[k]],1,0],{k,1,PrimePi[n-1]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A234695, A235330, A235508, A235592.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Jan 14 2014
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