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 A344174 #26 May 11 2021 10:32:18
%S A344174 0,1,1,1,1,2,1,1,2,2,1,3,1,3,1,2,2,3,2,1,2,2,2,1,3,1,3,4,4,4,3,4,2,4,
%T A344174 3,1,3,5,4,5,3,3,3,3,3,5,4,2,4,2,3,3,3,3,5,4,5,4,8,3,4,4,2,6,5,4,6,5,
%U A344174 8,3,4,3,5,3,3,6,4,6,4,4,4,3,5,4,7,6,3,6,5,4,7,4,5,2,4,6,2,4,6,7
%N A344174 Number of primes p of the form x^4 + y^2 with y + 1 prime such that p is smaller than prime(n) and also a quadratic nonresidue modulo prime(n).
%C A344174 In 1998 J. Friedlander and H. Iwaniec proved that there are infinitely many primes of the form x^4 + y^2 with x and y integers.
%C A344174 Conjecture: (i) a(n) > 0 for all n > 1. In other words, for each odd prime p, there is a prime q < p of the form x^4 + y^2 with y + 1 prime such that q is a quadratic nonresidue modulo p.
%C A344174 (ii) For any odd prime p not among 3, 5, 13, 37, 277, there is a prime q < p of the form x^4 + y^2 with y + 1 prime such that q is a quadratic residue modulo p.
%C A344174 Part (i) of the conjecture verified for all odd primes p < 2*10^9.
%C A344174 We even conjecture further that for any prime p > 5 there is a prime q < p of the form x^4 + y^2 with y + 1 prime such that q is a primitive root modulo p.
%C A344174 See also A344173 for a similar conjecture.
%H A344174 J. Friedlander and H. Iwaniec, <a href="https://arxiv.org/abs/math/9811185">The polynomial x^2 + y^4 captures its primes</a>, arXiv:math/9811185 [math.NT], 1998; Ann. of Math. 148 (1998), 945-1040.
%e A344174 a(2) = 1, and the prime 1^4 + (2-1)^2 = 2 is a quadratic nonresidue modulo prime(2) = 3.
%e A344174 a(4) = 1, and the prime 1^4 + (3-1)^2 = 5 is a quadratic nonresidue modulo prime(4) = 7.
%e A344174 a(24) = 1, and the prime 1^4 + (7-1)^2 = 37 is a quadratic nonresidue modulo prime(24) = 89.
%e A344174 a(36) = 1, and the prime 1^4 + (11-1)^2 = 101 is a quadratic nonresidue modulo prime(36) = 151.
%e A344174 a(204) = 5, and the primes 3^4 + (11-1)^2 = 181, 3^4 + (17-1)^2 = 337, 5^4 + (5-1)^2 = 641, 5^4 + (17-1)^2 = 881 and 5^4 + (23-1)^2 = 1109 are all quadratic nonresidues modulo prime(204) = 1249.
%t A344174 tab={0};Do[p:=p=Prime[n];tt={};Do[If[PrimeQ[b+1]&&PrimeQ[a^4+b^2]&&JacobiSymbol[a^4+b^2,p]==-1,tt=Append[tt,a^4+b^2]],{a,1,(p-1)^(1/4)},{b,1,(p-1-a^4)^(1/2)}];tab=Append[tab,Length[Union[tt]]],{n,2,100}];Print[tab]
%Y A344174 Cf. A000040, A000290, A000583, A028916, A281792, A344173.
%K A344174 nonn
%O A344174 1,6
%A A344174 _Zhi-Wei Sun_, May 10 2021