cp's OEIS Frontend

A219052 Number of ways to write n = p + q(3 - (-1)^n)/2 with q <= n/2 and p, q, p^2 + q^2 - 1 all prime.

%I A219052 #12 Apr 10 2013 19:16:19
%S A219052 0,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,1,1,0,1,2,1,1,0,2,1,1,0,2,2,0,2,1,
%T A219052 0,0,1,1,3,0,1,1,1,1,3,1,1,4,0,1,1,1,2,2,1,1,1,1,1,4,0,0,3,0,1,2,2,1,
%U A219052 3,1,2,3,2,1,3,2,4,2,1,2,1,1,0,4,2,1,1,1,2,5,4,1,3,1,1,4,1,1,2,2
%N A219052 Number of ways to write n = p + q(3 - (-1)^n)/2 with q <= n/2 and p, q, p^2 + q^2 - 1 all prime.
%C A219052 Conjecture: a(n) > 0 for all n > 784.
%C A219052 This conjecture implies Goldbach's conjecture, Lemoine's conjecture, and that there are infinitely many primes of the form p^2 + q^2 - 1 with p and q both prime.
%C A219052 It has been verified for n up to 10^8.
%C A219052 Zhi-Wei Sun also made the following general conjecture: Let d be any odd integer not congruent to 1 modulo 3. Then, all large even numbers can be written as p + q with p, q, p^2 + q^2 + d all prime. If d is also not divisible by 5, then all large odd numbers can be represented as p + 2q with p, q, p^2 + q^2 + d all prime.
%H A219052 Zhi-Wei Sun, <a href="/A219052/b219052.txt">Table of n, a(n) for n = 1..20000</a>
%H A219052 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv preprint arXiv:1211.1588, 2012.
%e A219052 a(12) = 1 since {5, 7} is the only prime pair {p, q} for which  p + q = 12, and p^2 + q^2 - 1 is prime.
%t A219052 a[n_] := a[n] = Sum[If[PrimeQ[n - (1 + Mod[n, 2])Prime[k]] == True && PrimeQ[Prime[k]^2 + (n - (1 + Mod[n, 2])Prime[k])^2 - 1] == True, 1, 0], {k, 1, PrimePi[n/2]}]; Do[Print[n, " ", a[n]], {n, 1, 20000}]
%Y A219052 Cf. A000040, A002375, A046927, A218754, A218585, A218654, A218825, A219023, A219026.
%K A219052 nonn,nice
%O A219052 1,22
%A A219052 _Zhi-Wei Sun_, Nov 10 2012