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%I #14 Sep 22 2023 11:02:37
%S 0,0,0,0,0,0,1,1,2,1,2,2,2,3,2,3,3,2,2,3,2,3,3,3,2,2,2,4,3,4,2,2,3,2,
%T 2,4,2,3,4,4,3,3,3,3,4,5,4,4,3,3,5,7,5,6,5,5,5,6,3,5,5,5,5,6,4,4,4,5,
%U 6,7,5,6,4,3,5,7,5,5,7,7,6,7,4,6,6,7,7,6,4,6,4,4,8,8,6,6,7,6,6,10
%N Number of ways to write n = (2-(n mod 2))*p + q + r with p <= q <= r such that p, q, r, p^2 - 2, q^2 - 2, r^2 - 2 are all prime.
%C Conjecture: a(n) > 0 for all n > 6.
%C This is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013. It also implies that there are infinitely many primes p with p^2 - 2 also prime.
%C Conjecture verified for n up to 10^9. - _Mauro Fiorentini_, Sep 22 2023
%H Zhi-Wei Sun, <a href="/A230502/b230502.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(10) = 1 since 10 = 2*2 + 3 + 3 with 2, 3, 2^2 - 2 = 2, 3^2 - 2 = 7 all prime.
%e a(19) = 2 since 19 = 3 + 3 + 13 = 5 + 7 + 7 with 3, 13, 5, 7, 3^2 - 2 = 7, 13^2 - 2 = 167, 5^2 - 2 = 23, 7^2 - 2 = 47 all prime.
%t pp[n_]:=PrimeQ[n^2-2]
%t pq[n_]:=PrimeQ[n]&&pp[n]
%t a[n_]:=Sum[If[pp[Prime[i]]&&pp[Prime[j]]&&pq[n-(2-Mod[n,2])Prime[i]-Prime[j]],1,0],{i,1,PrimePi[n/(4-Mod[n,2])]},{j,i,PrimePi[(n-(2-Mod[n,2])Prime[i])/2]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A062326, A068307, A230219, A230351, A230493, A230494.
%K nonn
%O 1,9
%A _Zhi-Wei Sun_, Oct 21 2013
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