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