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%I #21 Aug 04 2021 03:15:10
%S 5,2,19,127,17,67,163,41,89,101,131,313,257,211,227,461,241,401,613,
%T 337,433,353,577,467,863,887,617,787,601,569,761,641,823,673,857,1217,
%U 881,1091,1289,977,1427,1097,1801,929,1153,953,1321,1049,1747,1409
%N Least prime p with A271518(p) = n.
%C Conjecture: a(n) exists for any positive integer n.
%C In contrast, it is known that for each prime p the number of ordered integral solutions to the equation x^2 + y^2 + z^2 + w^2 = p is 8*(p+1).
%C In 1998 J. Friedlander and H. Iwaniec proved that there are infinitely many primes p of the form w^2 + x^4 = w^2 + (x^2)^2 + 0^2 + 0^2 with w and x nonnegative integers. Since x^2 + 3*0 + 5*0 is a square, we see that A271518(p) > 0 for infinitely many primes p.
%H Zhi-Wei Sun, <a href="/A276533/b276533.txt">Table of n, a(n) for n = 1..500</a>
%H 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.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.GM], 2016.
%e a(1) = 5 since 5 is the first prime which can be written in a unique way as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integer and x + 3*y + 5*z a square; in fact, 5 = 1^2 + 0^2 + 0^2 + 2^2 with 1 + 3*0 + 5*0 = 1^2.
%e a(2) = 2 since 2 = 1^2 + 0^2 + 0^2 + 1^2 with 1 + 3*0 + 5*0 = 1^2, and 2 = 1^2 + 1^2 + 0^2 + 0^2 with 1 + 3*1 + 5*0 = 2^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t Do[m=0;Label[aa];m=m+1;r=0;Do[If[SQ[Prime[m]-x^2-y^2-z^2]&&SQ[x+3y+5z],r=r+1;If[r>n,Goto[aa]]],{x,0,Sqrt[Prime[m]]},{y,0,Sqrt[Prime[m]-x^2]},{z,0,Sqrt[Prime[m]-x^2-y^2]}];If[r<n,Goto[aa],Print[n," ",Prime[m]]];Continue,{n,1,50}]
%Y Cf. A000040, A000118, A000290, A028916, A271518, A273294, A273302, A278560.
%K nonn
%O 1,1
%A _Zhi-Wei Sun_, Dec 12 2016
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