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A230230 Number of ways to write 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime.

%I A230230 #16 Jul 08 2023 16:30:35
%S A230230 0,0,0,1,1,1,2,2,2,2,3,4,3,3,5,1,5,5,3,4,5,3,2,6,4,3,6,3,3,6,3,5,6,3,
%T A230230 6,5,4,4,9,5,4,9,5,3,9,4,4,6,4,5,6,5,5,10,4,8,10,3,7,12,3,6,11,5,7,8,
%U A230230 3,4,6,6,4,7,2,7,9,2,10,9,3,9,8,3,5,14,8,4,12,5,5,11,5,6,8,3,8,7,4,9,11,3
%N A230230 Number of ways to write 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime.
%C A230230 Conjecture: a(n) > 0 for all n > 3.
%C A230230 This is stronger than Goldbach's conjecture for even numbers. If 2*n = p + q with p, q, 3*p - 10, 3*q + 10 all prime, then 6*n is the sum of the two primes 3*p - 10 and 3*q + 10.
%C A230230 Conjecture verified for 2*n up to 10^9. - _Mauro Fiorentini_, Jul 08 2023
%H A230230 Zhi-Wei Sun, <a href="/A230230/b230230.txt">Table of n, a(n) for n = 1..10000</a>
%H A230230 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588 [math.NT], 2012-2017.
%e A230230 a(5) = 1 since 2*5 = 7 + 3 with 3*7 - 10 = 11 and 3*3 + 10 = 19 both prime.
%e A230230 a(16) = 1 since 2*16 = 13 + 19 with 3*13 - 10 = 29 and 3*19 + 10 = 67 both prime.
%t A230230 PQ[n_]:=n>3&&PrimeQ[3n-10]
%t A230230 SQ[n_]:=PrimeQ[n]&&PrimeQ[3n+10]
%t A230230 a[n_]:=Sum[If[PQ[Prime[i]]&&SQ[2n-Prime[i]],1,0],{i,1,PrimePi[2n-2]}]
%t A230230 Table[a[n],{n,1,100}]
%Y A230230 Cf. A002375, A187757, A199920, A227923, A230227, A230140, A230141, A230217, A230219, A230223, A230224.
%K A230230 nonn
%O A230230 1,7
%A A230230 _Zhi-Wei Sun_, Oct 12 2013