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

1,1

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.

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.

Formula

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

Example

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

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]]

Prog

(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

Crossrefs

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

Sequence in context: A192419 A376757 A081836 * A351259 A154290 A267259

Adjacent sequences: A263718 A263719 A263720 * A263722 A263723 A263724

Keyword

nonn

Author

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

Status

approved