OFFSET

1,1

COMMENTS

Conjecture: All terms are odd. An equivalent conjecture in A263977 is that if k > 0 is even, then k^2 + p^2 is prime for some prime p. We have checked this for all k <= 12*10^7.

An odd number k is a term if and only if k^2 + 2^2 is composite, since k^2 + p^2 is even and > 2 (hence composite) for all odd primes p. Thus if the Conjecture is true, then the sequence is simply odd k such that k^2 + 4 is composite. The sequence of odd k, such that k^2 + 4 is prime, is A007591.

LINKS

Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, arXiv:math/0703284 [math.NT], 2007-2008; Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.

Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

EXAMPLE

9^2 + 2^2 = 85 = 5*17, 11^2 + 2^2 = 125 = 5^3, and 23^2 + 2^2 = 533 = 13*41 are composite, so 9, 11, and 23 are members.

1^2 + 2^2 = 5 and 2^2 + 3^3 = 13 are prime, so 1, 2, and 3 are not members.

CROSSREFS

KEYWORD

nonn

AUTHOR

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

STATUS

approved