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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.

%I A287930 #20 Jul 13 2017 20:42:10
%S A287930 1,3,21,33,93,105,123,177,219,237,321,417,489,537,633,699,813,951,
%T A287930 1011,1299,1419,1641,1923,1959,2073,2211,2433,2661,3387,3453,3489,
%U A287930 3741,3981,4083,4377,4461,4467,4827,4911,5007,5997,6423,6621,7467,7647,7881,8031,8061
%N 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.
%C A287930 The sequence contains A287799.
%C A287930 a(n) == 3 or 15 (mod 18) for n > 1.
%C A287930 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.
%C A287930 From _Robert Israel_, Jul 13 2017: (Start)
%C A287930 Not all b(m) for m > 1 are semiprime.
%C A287930 A counterexample is a(8821) = 23963385 = 3*5*373*4283.
%C A287930 All terms are squarefree. (End)
%H A287930 Robert Israel, <a href="/A287930/b287930.txt">Table of n, a(n) for n = 1..10000</a>
%e A287930 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.
%p A287930 filter:= proc(m)
%p A287930    andmap(x -> isprime(x^2 + 2*(m/x)^2),
%p A287930      select(t -> t^2 <= m,numtheory:-divisors(m)));
%p A287930 end proc:
%p A287930 select(filter, [1, seq(i,i=3..10000,3)]); # _Robert Israel_, Jul 13 2017
%t A287930 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
%Y A287930 Cf. A000040, A287799.
%K A287930 nonn
%O A287930 1,2
%A A287930 _Michel Lagneau_, Jun 03 2017
%E A287930 Edited by _Robert Israel_, Jul 13 2017