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

1,2

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. - Robert Israel, Jul 14 2017

Links

Robert Israel, Table of n, a(n) for n = 1..300

Wikipedia, Bunyakovsky conjecture.

Example

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.

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

Prog

(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

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.

Sequence in context: A039766 A072849 A287930 * A089323 A100986 A213141

Adjacent sequences: A287796 A287797 A287798 * A287800 A287801 A287802

Keyword

nonn

Author

Michel Lagneau, Jun 01 2017

Extensions

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

Status

approved