A263721 The prime p in the Fouvry-Iwaniec prime k^2 + p^2 (A185086), or the larger of k and p if both are prime.

2, 3, 5, 5, 7, 5, 3, 5, 3, 7, 11, 7, 11, 13, 7, 2, 13, 13, 5, 17, 13, 11, 5, 17, 7, 17, 19, 3, 17, 7, 19, 5, 11, 19, 13, 23, 5, 17, 19, 13, 23, 5, 2, 19, 17, 11, 5, 23, 29, 29, 23, 19, 29, 13, 31, 31, 23, 11, 3, 5, 31, 13, 2, 29, 5, 13, 31, 2, 11, 5, 31, 37, 23, 37, 3, 7, 23, 3, 13, 31, 19, 37, 41, 11

Offset: 1

Author

Jonathan Sondow and Robert G. Wilson v, Oct 24 2015

Keywords

• nonn

Comments

The sequence is well-defined by the uniqueness part of Fermat's two-squares theorem.

The sequence is infinite, since Fouvry and Iwaniec proved that A185086 is infinite.

Examples

			A185086(2) = 13 = 2^2 + 3^2 and A185086(6) = 61 = 5^2 + 6^2, so a(2) = 3 and a(6) = 5.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Étienne Fouvry and Henryk Iwaniec, Gaussian primes, Acta Arithmetica 79:3 (1997), pp. 249-287.

Crossrefs

Cf. A002313, A002144, A028916, A045637, A062324, A185086, A240130, A262340, A263722.

Programs

Mathematica

p = 2; lst = {}; While[p < 100, k = 1; While[k < 101, If[PrimeQ[k^2 + p^2], AppendTo[lst, {k^2 + p^2, If[PrimeQ@ k, Max[k, p], p]}]]; k++]; p = NextPrime@ p]; Transpose[Union@ lst][[2]]

PARI

do(lim)=my(v=List(),p2,t); forprime(p=2, sqrtint(lim\=1), p2=p^2; for(k=1, sqrtint(lim-p2), if(isprime(t=p2+k^2), listput(v, [t, if(isprime(k),max(k,p),p)])))); v=vecsort(Set(v),1); apply(u->u[2], v) \\ Charles R Greathouse IV, Aug 21 2017

Formula

a(n)^2 = A185086(n) - k^2 for some integer k > 0.