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A263722 Integers k > 0 such that k^2 + p^2 is composite for all primes p. 5
9, 11, 19, 21, 23, 25, 29, 31, 39, 41, 43, 49, 51, 53, 55, 59, 61, 63, 69, 71, 75, 77, 79, 81, 83, 89, 91, 93, 99, 101, 105, 107, 109, 111, 113, 119, 121, 123, 127, 129, 131, 133, 139, 141, 143, 145, 149, 151, 153, 157, 159, 161, 165, 169, 171, 173, 175, 179, 181, 185, 187, 189, 191, 195, 197, 199 (list; graph; refs; listen; history; text; internal format)
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.

The complementary sequence is A263977. Given k in it, the smallest prime p, such that k^2 + p^2 is prime, is in A263726. These numbers k^2 + p^2 form A185086, the Fouvry-Iwaniec primes.

LINKS

Table of n, a(n) for n=1..66.

Stephan Baier and Liangyi Zhao, On Primes Represented by Quadratic Polynomials, Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.

√Čtienne 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.

MATHEMATICA

Select[1 + 2Range@ 99, !PrimeQ[#^2 + 4] &] (* assumes the Conjecture *)

CROSSREFS

Cf. A002313, A045637, A062324, A185086, A007591, A240130, A240131, A263466, A263726, A263977.

Sequence in context: A077788 A137018 A182391 * A299971 A090771 A284295

Adjacent sequences:  A263719 A263720 A263721 * A263723 A263724 A263725

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified February 23 09:04 EST 2019. Contains 320420 sequences. (Running on oeis4.)