cp's OEIS Frontend

A227908 Number of ways to write 2*n = p + q with p, q and (p-1)^2 + q^2 all prime.

%I A227908 #20 Sep 21 2023 15:36:23
%S A227908 0,1,1,1,2,1,1,2,2,2,1,2,3,2,2,0,2,6,1,3,5,2,3,2,1,2,2,5,4,3,2,3,8,1,
%T A227908 4,3,3,2,5,1,2,4,5,3,4,4,2,6,1,4,5,3,3,6,2,6,5,4,5,7,3,1,9,2,3,6,1,2,
%U A227908 5,4,7,2,7,6,6,2,4,10,3,3,6,1,7,9,5,4,5,4,3,5,3,5,8,4,4,5,2,11,9,4
%N A227908 Number of ways to write 2*n = p + q with p, q and (p-1)^2 + q^2 all prime.
%C A227908 Conjecture: a(n) > 0 except for n = 1, 16, 292.
%C A227908 This implies not only Goldbach's conjecture for even numbers, but also Ming-Zhi Zhang's conjecture (cf. A036468) that any odd number greater than one can be written as x + y (x, y > 0) with x^2 + y^2 prime.
%C A227908 We have verified the conjecture for n up to 10^7.
%C A227908 Conjecture verified for n up tp 10^9. - _Mauro Fiorentini_, Sep 21 2023
%H A227908 Zhi-Wei Sun, <a href="/A227908/b227908.txt">Table of n, a(n) for n = 1..10000</a>
%H A227908 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2017.
%e A227908 a(7) = 1 since 2*7 = 11 + 3, and (11-1)^2 + 3^2 = 109 is prime.
%e A227908 a(19) = 1 since 2*19 = 7 + 31, and (7-1)^2 + 31^2 = 997 is prime.
%t A227908 a[n_]:=Sum[If[PrimeQ[2n-Prime[i]]&&PrimeQ[(Prime[i]-1)^2+(2n-Prime[i])^2],1,0],{i,1,PrimePi[2n-2]}]
%t A227908 Table[a[n],{n,1,100}]
%Y A227908 Cf. A002375, A036468, A220554, A230224.
%K A227908 nonn
%O A227908 1,5
%A A227908 _Zhi-Wei Sun_, Oct 12 2013