A287799 -id:A287799 - cp's OEIS frontend

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

Michel Lagneau, Jun 03 2017

nonn

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)

			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.

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

Cf. A000040, A287799.

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

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

Edited by Robert Israel, Jul 13 2017

cp's OEIS Frontend

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.

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