A287930 Numbers m such that for any positive integers (x, y), if x * y = m where x <= y, then x^2 + 2*y^2 is a prime number.

1, 3, 21, 33, 93, 105, 123, 177, 219, 237, 321, 417, 489, 537, 633, 699, 813, 951, 1011, 1299, 1419, 1641, 1923, 1959, 2073, 2211, 2433, 2661, 3387, 3453, 3489, 3741, 3981, 4083, 4377, 4461, 4467, 4827, 4911, 5007, 5997, 6423, 6621, 7467, 7647, 7881, 8031, 8061

Offset

1,2

Comments

The sequence contains A287799.

a(n) == 3 or 15 (mod 18) for n > 1.

The numbers a(n)/3 are 1, 7, 11, 31, 35, 41, 59, 73, 79, 107, ... with a majority of prime numbers, except the subset {b(m)} = {1, 35, 473, 737, 1247, 2489, 2627, ...}. It seems that b(m) is semiprime for m > 1.

From Robert Israel, Jul 13 2017: (Start)

Not all b(m) for m > 1 are semiprime.

A counterexample is a(8821) = 23963385 = 3*5*373*4283.

All terms are squarefree. (End)

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

105 = 1*105 = 3*35 = 5*21 = 7*15 => 1^2 + 2*105^2 = 22051, 3^2 + 2*35^2 = 2459, 5^2 + 2*21^2 = 907 and 7^2 + 2*15^2 = 499 are primes.

Maple

filter:= proc(m)
   andmap(x -> isprime(x^2 + 2*(m/x)^2),
     select(t -> t^2 <= m, numtheory:-divisors(m)));
end proc:
select(filter, [1, seq(i, i=3..10000, 3)]); # Robert Israel, Jul 13 2017

Mathematica

t={}; Do[ds=Divisors[n]; If[EvenQ[Length[ds]], ok=True; k=1; While[k<=Length[ds]/2&&(ok=PrimeQ[ds[[k]]^2+2*ds[[-k]]^2]), k++]; If[ok, AppendTo[t, n]]], {n, 2, 10^4}]; t

Crossrefs

Cf. A000040, A287799.

Sequence in context: A317860 A039766 A072849 * A287799 A089323 A100986

Adjacent sequences: A287927 A287928 A287929 * A287931 A287932 A287933

Keyword

nonn

Author

Michel Lagneau, Jun 03 2017

Extensions

Edited by Robert Israel, Jul 13 2017

Status

approved