A287799 If x^2 + 2*y^2 is prime for all positive integers x and y with m = x*y then m is in the sequence.

1, 3, 21, 33, 123, 219, 321, 3453, 6621, 16521, 18273, 25089, 27831, 29787, 62313, 69981, 75459, 95577, 101301, 105459, 157299, 196239, 197481, 247047, 259797, 281433, 359943, 390237, 418881, 460821, 529167, 569559, 595869, 680307, 727341, 945141, 955569, 964401

Offset: 1

Author

Michel Lagneau, Jun 01 2017

Keywords

• nonn

Comments

a(n) == 3, 15 (mod 18), for n >= 2.

It seems that a(n) = 3*p where p is a prime, for n >= 3.

a(n) is squarefree. - David A. Corneth, Jun 01 2017

From Robert Israel, Jul 14 2017: (Start)

a(149) = 13304379 = 3*11*403163 is not of the form 3*p.

The generalized Bunyakovsky conjecture implies that there are infinitely many terms of the form 3*p, and infinitely many of the form 3*11*p. (End)

Examples

			1 = 1*1 and 1^2 + 2*1^2 = 3, a prime.
21 = 1*21 = 3*7 = 21*1 = 7*3 => 1^2 + 2*21^2 = 883, 3^2 + 2*7^2 = 107, 21^2 + 2*1^2 = 443 and 7^2 + 2*3^2 = 67 are primes.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..300
• Wikipedia, Bunyakovsky conjecture.

Crossrefs

Cf. A000040, A005117, A236423 (with x^2+y^2 instead of x^2+2*y^2).

Subsequence of A067201. - Michel Marcus, Jun 03 2017

Cf. A002332, A002333.

Programs

Maple

filter:= proc(m)
   andmap(x -> isprime(x^2 + 2*(m/x)^2),
  numtheory:-divisors(m));
end proc:
select(filter, [1, seq(seq(18*i+j, j=[3,15]),i=0..10^5)]); # Robert Israel, Jul 14 2017

Mathematica

A287799 = {}; Do[ds = Divisors[n]; If[EvenQ[Length[ds]], flag = True; k = 1; While[k <= Length[ds]/2 && (criterion1 = PrimeQ[ds[[k]]^2 + 2 * ds[[-k]]^2]) && (criterion2 = PrimeQ[ds[[-k]]^2 + 2 * ds[[k]]^2]), k++]; If[criterion1 && criterion2, AppendTo[A287799, n]]], {n, 2, 10^6}]; A287799

PARI

is(n) = d=divisors(n); for(i=1,#d, if(!isprime(d[i]^2 + 2*d[#d-i+1]^2), return(0))); n > 1 \\ David A. Corneth, Jun 01 2017

Sage

R = range(1,100000)
[m for m in R if all(is_prime(d^2+2*(m//d)^2) for d in divisors(m))] # Peter Luschny, Jun 18 2017

Extensions

Name reformulated and m=1 added by Wolfdieter Lang, Jun 20 2017