This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A240130 #17 Feb 16 2025 08:33:21
%S A240130 5,13,29,53,137,173,293,397,593,857,977,1373,1697,1913,2213,2909,3517,
%T A240130 3821,4493,5077,5333,6257,7213,7937,9413,10301,10613,11549,11897,
%U A240130 13093,16193,17417,18773,19421,22397,22817,24749,26573,27893,30029
%N A240130 Least prime of the form prime(n)^2 + k^2, or 0 if none.
%C A240130 The positive terms form a subsequence of A185086 = Fouvry-Iwaniec primes = primes of the form prime^2 + integer^2.
%C A240130 The values of k are A240131.
%C A240130 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 A240130 Robert Israel, <a href="/A240130/b240130.txt">Table of n, a(n) for n = 1..10000</a>
%H A240130 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 A240130 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 A240130 E.W. Weisstein, <a href="https://mathworld.wolfram.com/Fermats4nPlus1Theorem.html">Fermat's 4n+1 Theorem</a>, MathWorld.
%H A240130 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bunyakovsky_conjecture">Bunyakovsky's conjecture</a>
%F A240130 a(n) == 1 (mod 4) if a(n) > 0.
%F A240130 a(n) > 0 if Bunyakovsky's conjecture is true.
%F A240130 a(n) <> a(m) if n <> m and a(n) > 0, by uniqueness in Fermat's 4n+1 Theorem.
%F A240130 a(n) = prime(n)^2 + A240131(n)^2 if a(n) > 0.
%e A240130 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 A240130 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 A240130 g(2):= 5:
%p A240130 seq(g(ithprime(i)),i=1..1000); # _Robert Israel_, Nov 04 2015
%t A240130 Table[First[Select[Prime[n]^2 + Range[20]^2, PrimeQ]], {n, 40}]
%o A240130 (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 A240130 Cf. A002144, A185086.
%K A240130 nonn
%O A240130 1,1
%A A240130 _Jonathan Sondow_, Apr 07 2014