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A236423 Numbers k such that m^2 + k^2/m^2 is prime for every m|k.

%I A236423 #26 Jun 28 2025 17:57:11
%S A236423 1,2,6,10,14,26,74,94,130,134,146,170,206,326,386,466,470,634,1094,
%T A236423 1354,1570,1654,1766,1966,2174,2766,3046,3254,3274,3446,4006,4174,
%U A236423 4666,4754,4954,5086,5774,5834,6046,6866,6926,7114,7466,8854,9046,9494,10006,10126
%N A236423 Numbers k such that m^2 + k^2/m^2 is prime for every m|k.
%C A236423 If n = x*y then x^2 + y^2 is a prime.
%C A236423 These n > 1 must be even and squarefree.
%C A236423 Conjecture: the set of such n is infinite.
%C A236423 The conjecture follows from, e.g., Schinzel's hypothesis H. - _Charles R Greathouse IV_, Jan 28 2014
%H A236423 Giovanni Resta, <a href="/A236423/b236423.txt">Table of n, a(n) for n = 1..1000</a>
%t A236423 Select[Range[10^4], (d = Divisors[#]^2; And @@ PrimeQ[d + #^2/d]) &] (* _Giovanni Resta_, Jan 26 2014 *)
%o A236423 (PARI) isok(n) = sumdiv(n, d, isprime(d^2 + n^2/d^2)) == numdiv(n); \\ _Michel Marcus_, Jan 25 2014
%o A236423 (PARI) is(n)=if(n%4!=2, return(n==1)); my(f=factor(n)); if(vecmax(f[,2])>1,return(0)); fordiv(f,m,if(!isprime(m^2+(n/m)^2),return(0)); if(m^2>n,break));1 \\ _Charles R Greathouse IV_, Jan 28 2014
%Y A236423 Cf. A080715.
%Y A236423 Subsequence of A005574. - _Michel Marcus_, Jun 03 2017
%K A236423 nonn
%O A236423 1,2
%A A236423 _Thomas Ordowski_, Jan 25 2014
%E A236423 More terms from _Michel Marcus_, Jan 25 2014