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%I #18 Aug 07 2023 07:59:00
%S 0,0,0,1,2,2,1,1,3,3,2,1,4,3,4,2,4,3,4,5,4,2,3,6,3,3,3,5,2,3,3,3,1,2,
%T 4,2,2,3,3,1,5,2,3,3,7,3,5,4,6,3,5,6,5,5,3,6,2,5,5,3,4,5,6,2,6,6,5,1,
%U 5,3,3,3,2,2,5,6,5,1,5,6,4,4,6,6,1,5,5,4,3,4,3,3,6,5,4,1,5,7,2,4
%N Number of ordered ways to write n = p + q (q > 0) with p, 2*p^2 - 1 and 2*q^2 - 1 all prime.
%C Conjecture: a(n) > 0 for all n > 3.
%C We have verified this for n up to 2*10^7.
%C Conjecture verified for n up to 10^9. - _Mauro Fiorentini_, Aug 07 2023
%H Zhi-Wei Sun, <a href="/A230351/b230351.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 a(7) = 1 since 7 = 3 + 4 with 3, 2*3^2 - 1 = 17, 2*4^2 - 1 = 31 all prime.
%e a(40) = 1 since 40 = 2 + 38, and 2, 2*2^2 - 1 = 7 , 2*38^2 - 1 = 2887 are all prime.
%e a(68) = 1 since 68 = 43 + 25, and all the three numbers 43, 2*43^2 - 1 = 3697 and 2*25^2 - 1 = 1249 are prime.
%t a[n_]:=Sum[If[PrimeQ[2Prime[i]^2-1]&&PrimeQ[2(n-Prime[i])^2-1],1,0],{i,1,PrimePi[n-1]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A066049, A106483, A219864, A230252, A230254, A230261.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Oct 16 2013
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