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%I #13 Aug 29 2017 14:19:30
%S 0,1,1,0,1,3,1,0,1,2,2,0,3,7,3,0,4,4,1,0,4,7,3,0,3,5,2,0,4,6,2,0,2,3,
%T 3,0,4,8,3,0,5,8,2,0,2,5,2,0,5,8,4,0,4,5,2,0,5,6,4,0,1,8,5,0,3,9,3,0,
%U 6,8,3,0,5,13,5,0,9,9,2,0,4,6,6,0,7,11,4,0,8,10,5,0,2,11,5,0,3,10,4,0
%N Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that p = x + 2*y + 5*z, p - 2 and p + 4 are all prime.
%C Conjecture: a(n) > 0 for all n > 1 not divisible by 4.
%C See also A291635 for a stronger conjecture.
%H Zhi-Wei Sun, <a href="/A291624/b291624.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.
%e a(2) = 1 since 2 = 0^2 + 1^2 + 1^2 + 0^2 with 0 + 2*1 + 5*1 = 7, 7 - 2 = 5 and 7 + 4 = 11 all prime.
%e a(5) = 1 since 5 = 2^2 + 0^2 + 1^2 + 0^2 with 2 + 2*0 + 5*1 = 7, 7 - 2 = 5 and 7 + 4 = 11 all prime.
%e a(181) = 1 since 181 = 1^2 + 6^2 + 0^2 + 12^2 with 1 + 2*6 + 5*0 = 13, 13 - 2 = 11 and 13 + 4 = 17 all prime.
%e a(285) = 1 since 285 = 10^2 + 4^2 + 5^2 + 12^2 with 10 + 2*4 + 5*5 = 43, 43 - 2 = 41 and 43 + 4 = 47 all prime.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t TQ[p_]:=TQ[p]=PrimeQ[p]&&PrimeQ[p-2]&&PrimeQ[p+4];
%t Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&TQ[x+2y+5z],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r],{n,1,100}]
%Y Cf. A000040, A000118, A000290, A022004, A271518, A281976, A290935, A291150, A291191, A291455, A291635.
%K nonn,look
%O 1,6
%A _Zhi-Wei Sun_, Aug 28 2017
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