A240130 - OEIS

%I #17 Feb 16 2025 08:33:21

%S 5,13,29,53,137,173,293,397,593,857,977,1373,1697,1913,2213,2909,3517,

%T 3821,4493,5077,5333,6257,7213,7937,9413,10301,10613,11549,11897,

%U 13093,16193,17417,18773,19421,22397,22817,24749,26573,27893,30029

%N Least prime of the form prime(n)^2 + k^2, or 0 if none.

%C The positive terms form a subsequence of A185086 = Fouvry-Iwaniec primes = primes of the form prime^2 + integer^2.

%C The values of k are A240131.

%C Is a(n) < a(n+1) for all n? (I have checked it for n <= 10^6.) Note that A240131 is far from being monotone.

%H Robert Israel, <a href="/A240130/b240130.txt">Table of n, a(n) for n = 1..10000</a>

%H Stephan Baier and Liangyi Zhao, <a href="http://arxiv.org/abs/math/0703284">On Primes Represented by Quadratic Polynomials</a>, Anatomy of Integers, CRM Proc. & Lecture Notes, Vol. 46, Amer. Math. Soc. 2008, pp. 169 - 166.

%H Étienne Fouvry and Henryk Iwaniec, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa79/aa7935.pdf">Gaussian primes</a>, Acta Arithmetica 79:3 (1997), pp. 249-287.

%H E.W. Weisstein, <a href="https://mathworld.wolfram.com/Fermats4nPlus1Theorem.html">Fermat's 4n+1 Theorem</a>, MathWorld.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bunyakovsky_conjecture">Bunyakovsky's conjecture</a>

%F a(n) == 1 (mod 4) if a(n) > 0.

%F a(n) > 0 if Bunyakovsky's conjecture is true.

%F a(n) <> a(m) if n <> m and a(n) > 0, by uniqueness in Fermat's 4n+1 Theorem.

%F a(n) = prime(n)^2 + A240131(n)^2 if a(n) > 0.

%e Prime(2) = 3 and 3^2 + 1^2 = 10 is not prime but 3^2 + 2^2 = 13 is prime, so a(2) = 13.

%p g:= proc(p) local k; for k from 2 by 2 do if isprime(p^2 + k^2) then return p^2+k^2 fi od end proc:

%p g(2):= 5:

%p seq(g(ithprime(i)),i=1..1000); # _Robert Israel_, Nov 04 2015

%t Table[First[Select[Prime[n]^2 + Range[20]^2, PrimeQ]], {n, 40}]

%o (PARI) a(n) = {p = prime(n); k = 1 - p%2; inc = 2; while (!isprime(q=p^2+k^2), k += inc); q;} \\ _Michel Marcus_, Nov 04 2015

%Y Cf. A002144, A185086.

%K nonn

%O 1,1

%A _Jonathan Sondow_, Apr 07 2014