A230494 Number of ways to write n = x^2 + y (x, y >= 0) with 2*y^2 - 1 prime.

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, 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, 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

Offset: 1

Author

Zhi-Wei Sun, Oct 20 2013

Keywords

• nonn

Comments

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.

(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.

We have verified the conjecture for n up to 2*10^7.

Both conjectures verified for n up to 10^9. - Mauro Fiorentini, Aug 08 2023

See also A230351 and A230493 for similar conjectures.

Examples

			a(9) = 1 since 9 = 1^2 + 8 with 2*8^2 - 1 = 127 prime.
a(69) = 1 since 69 = 0^2 + 69 with 2*69^2 - 1 = 9521 prime.
a(76) = 1 since 76 = 0^2 + 76 with 2*76^2 - 1 = 11551 prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Crossrefs

Cf. A000040, A000290, A066049, A220272, A229166, A230351, A230493.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[2(n-x^2)^2-1],1,0],{x,0,Sqrt[n]}]
Table[a[n],{n,1,100}]