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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

√Čtienne 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: A326399 A192419 A081836 * A154290 A267259 A002334

Adjacent sequences:  A263718 A263719 A263720 * A263722 A263723 A263724

KEYWORD

nonn

AUTHOR

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

STATUS

approved

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Last modified August 3 20:08 EDT 2020. Contains 336201 sequences. (Running on oeis4.)