%I #15 Aug 08 2023 12:52:04
%S 0,1,2,2,1,2,3,3,1,2,4,3,2,2,3,2,3,3,4,2,2,5,2,3,3,4,3,3,4,1,3,2,3,3,
%T 2,2,3,5,3,5,2,5,6,3,3,5,5,1,4,6,4,4,5,4,3,3,4,3,5,4,4,3,4,5,3,5,4,5,
%U 1,5,4,4,4,5,4,1,6,3,3,3,5,4,2,3,8,3,4,6,6,2,4,7,1,4,4,5,1,6,5,3
%N Number of ways to write n = x^2 + y (x, y >= 0) with 2*y^2 - 1 prime.
%C Conjecture: (i) a(n) > 0 for all n > 1. Moreover, if n > 1 is not among 2, 69, 76, then there are positive integers x and y such that x^2 + y is equal to n and 2*y^2 - 1 is prime.
%C (ii) Any integer n > 1 can be written as x*(x+1)/2 + y with 2*y^2 - 1 prime, where x and y are nonnegative integers. Moreover, if n is not equal to 2 or 15, then we may require additionally that x and y are both positive.
%C We have verified the conjecture for n up to 2*10^7.
%C Both conjectures verified for n up to 10^9. - _Mauro Fiorentini_, Aug 08 2023
%C See also A230351 and A230493 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A230494/b230494.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.
%e a(9) = 1 since 9 = 1^2 + 8 with 2*8^2 - 1 = 127 prime.
%e a(69) = 1 since 69 = 0^2 + 69 with 2*69^2 - 1 = 9521 prime.
%e a(76) = 1 since 76 = 0^2 + 76 with 2*76^2 - 1 = 11551 prime.
%t a[n_]:=Sum[If[PrimeQ[2(n-x^2)^2-1],1,0],{x,0,Sqrt[n]}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000290, A066049, A220272, A229166, A230351, A230493.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Oct 20 2013
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