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

1, 2, 6, 10, 14, 26, 74, 94, 130, 134, 146, 170, 206, 326, 386, 466, 470, 634, 1094, 1354, 1570, 1654, 1766, 1966, 2174, 2766, 3046, 3254, 3274, 3446, 4006, 4174, 4666, 4754, 4954, 5086, 5774, 5834, 6046, 6866, 6926, 7114, 7466, 8854, 9046, 9494, 10006, 10126

Offset: 1

Author

Thomas Ordowski, Jan 25 2014

Keywords

• nonn

Comments

If n = x*y then x^2 + y^2 is a prime.

These n > 1 must be even and squarefree.

Conjecture: the set of such n is infinite.

The conjecture follows from, e.g., Schinzel's hypothesis H. - Charles R Greathouse IV, Jan 28 2014

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A080715.

Subsequence of A005574. - Michel Marcus, Jun 03 2017

Programs

Mathematica

Select[Range[10^4], (d = Divisors[#]^2; And @@ PrimeQ[d + #^2/d]) &] (* Giovanni Resta, Jan 26 2014 *)

PARI

isok(n) = sumdiv(n, d, isprime(d^2 + n^2/d^2)) == numdiv(n); \\ Michel Marcus, Jan 25 2014

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

Extensions

More terms from Michel Marcus, Jan 25 2014