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A230516 Number of ways to write n = a + b + c with 0 < a <= b <= c such that {a^2+a-1, a^2+a+1}, {b^2+b-1, b^2+b+1}, {c^2+c-1, c^2+c+1} are twin prime pairs.

%I A230516 #15 Sep 22 2023 12:17:11
%S A230516 0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,2,3,3,2,2,2,2,3,4,4,3,5,5,4,6,6,4,4,
%T A230516 4,3,3,4,1,2,3,4,4,5,6,6,7,6,6,7,6,4,3,5,4,4,3,5,5,6,8,6,7,11,7,6,9,8,
%U A230516 4,8,6,5,7,5,4,8,10,5,7,9,6,10,6,7,7,7,4,4,8,5,5,4,6,9,7,7,7,7,7,8
%N A230516 Number of ways to write n = a + b + c with 0 < a <= b <= c such that {a^2+a-1, a^2+a+1}, {b^2+b-1, b^2+b+1}, {c^2+c-1, c^2+c+1} are twin prime pairs.
%C A230516 Conjecture: a(n) > 0 for all n > 5.
%C A230516 Conjecture verified for n up to 10^9. - _Mauro Fiorentini_, Sep 22 2023
%C A230516 This implies that there are infinitely many twin prime pairs of the form {x^2 + x - 1, x^2 + x + 1}.
%C A230516 See also A230514 for a similar conjecture.
%H A230516 Zhi-Wei Sun, <a href="/A230516/b230516.txt">Table of n, a(n) for n = 1..5000</a>
%H A230516 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 A230516 a(8) = 1 since 8 = 2 + 3 + 3, and {2*3 - 1, 2*3 + 1} = {5, 7} and {3*4 - 1, 3*4 + 1} = {11, 13} are twin prime pairs.
%e A230516 a(39) = 1 since 39 = 3 + 15 + 21, and {3*4 - 1, 3*4 + 1} = {11, 13}, {15*16 - 1, 15*16 + 1} = {239, 241}, {21*22 - 1, 21*22 + 1} = {461, 463} are twin prime pairs.
%t A230516 pp[n_]:=PrimeQ[n(n+1)-1]&&PrimeQ[n(n+1)+1]
%t A230516 a[n_]:=Sum[If[pp[i]&&pp[j]&&pp[n-i-j],1,0],{i,1,n/3},{j,i,(n-i)/2}]
%t A230516 Table[a[n],{n,1,100}]
%Y A230516 Cf. A001359, A006512, A088485, A227923, A230219, A230252, A230351, A230514.
%K A230516 nonn
%O A230516 1,9
%A A230516 _Zhi-Wei Sun_, Oct 22 2013