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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 3 14:34 EDT 2022. Contains 357237 sequences. (Running on oeis4.)