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A230494 Number of ways to write n = x^2 + y (x, y >= 0) with 2*y^2 - 1 prime. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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.

See also A230351 and A230493 for similar conjectures.

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.

EXAMPLE

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.

MATHEMATICA

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

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

CROSSREFS

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

Sequence in context: A294232 A331254 A193738 * A106580 A165915 A269783

Adjacent sequences:  A230491 A230492 A230493 * A230495 A230496 A230497

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Oct 20 2013

STATUS

approved

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Last modified October 22 18:45 EDT 2021. Contains 348175 sequences. (Running on oeis4.)