A227908 - OEIS

%I #20 Sep 21 2023 15:36:23

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

%C Conjecture: a(n) > 0 except for n = 1, 16, 292.

%C 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 We have verified the conjecture for n up to 10^7.

%C Conjecture verified for n up tp 10^9. - _Mauro Fiorentini_, Sep 21 2023

%H Zhi-Wei Sun, <a href="/A227908/b227908.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>, arXiv:1211.1588 [math.NT], 2012-2017.

%e a(7) = 1 since 2*7 = 11 + 3, and (11-1)^2 + 3^2 = 109 is prime.

%e a(19) = 1 since 2*19 = 7 + 31, and (7-1)^2 + 31^2 = 997 is prime.

%t 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 Table[a[n],{n,1,100}]

%Y Cf. A002375, A036468, A220554, A230224.

%K nonn

%O 1,5

%A _Zhi-Wei Sun_, Oct 12 2013