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 A344173 #26 Jun 26 2021 06:49:00
%S A344173 0,1,2,1,1,1,1,2,1,3,3,3,3,3,2,2,2,4,3,1,3,4,2,3,2,3,2,3,2,4,3,4,4,4,
%T A344173 3,1,3,5,2,4,2,3,2,3,3,3,5,1,3,3,4,1,3,4,3,2,4,4,4,2,4,4,4,3,3,5,3,3,
%U A344173 2,3,1,2,6,4,6,2,4,3,4,3,4,5,4,2,4,5,4,1,5,3,3,6,4,4,3,4,3,3,5,4
%N A344173 Number of primes p < prime(n) of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that p is a quadratic nonresidue modulo prime(n).
%C A344173 In 2001 Heath-Brown proved that there are infinitely many primes of the form x^3 + 2*y^3 with x and y nonnegative integers.
%C A344173 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^3 + 2*y^3 with x nonnegative and y + 1 prime such that q is a quadratic nonresidue modulo p.
%C A344173 (ii) For any prime p > 5 not equal to 29, there is a prime q < p of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that q is a quadratic residue modulo p.
%C A344173 Part (i) of the conjecture verified for all odd primes p < 2*10^9.
%C A344173 We even conjecture further that for any prime p > 5 there is a prime q < p of the form x^3 + 2*y^3 with x nonnegative and y + 1 prime such that q is a primitive root modulo p.
%C A344173 See also A344174 for a similar conjecture.
%H A344173 D. R. Heath-Brown, <a href="http://doi.org/10.1007/BF02392715">Primes represented by x^3 + 2y^3</a>, Acta Mathematica 186 (2001), 1-84.
%e A344173 a(2) = 1, and the prime 0^3 + 2*(2-1)^3 = 2 is a quadratic nonresidue modulo prime(2) = 3.
%e A344173 a(6) = 1, and the prime 0^3 + 2*(2-1)^3 = 2 is a quadratic nonresidue modulo prime(6) = 13.
%e A344173 a(20) = 1, and the prime 1^3 + 2*(2-1)^3 = 17 is a quadratic nonresidue modulo prime(20) = 71.
%e A344173 a(48) = 1, and the prime 1^3 + 2*(2-1)^3 = 3 is a quadratic nonresidue modulo prime(48) = 223.
%e A344173 a(88) = 1, and the prime 3^3 + 2*(3-1)^3 = 43 is a quadratic nonresidue modulo prime(88) = 457.
%t A344173 tab={0};Do[p:=p=Prime[n];tt={};Do[If[PrimeQ[b+1]&&PrimeQ[a^3+2b^3]&&JacobiSymbol[a^3+2b^3,p]==-1,tt=Append[tt,a^3+2b^3]],{a,0,(p-1)^(1/3)},{b,1,((p-1-a^3)/2)^(1/3)}];tab=Append[tab,Length[Union[tt]]],{n,2,100}];Print[tab]
%Y A344173 Cf. A000040, A000578, A173587, A220413, A344174.
%K A344173 nonn
%O A344173 1,3
%A A344173 _Zhi-Wei Sun_, May 10 2021