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A282545 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y^2 + 64*z^2 + 1024*y*z a square, where x,y,z,w are nonnegative integers with y > 0.

%I A282545 #10 Feb 18 2017 14:27:08
%S A282545 1,3,3,2,4,5,2,2,5,5,3,2,4,5,2,1,6,9,5,5,8,6,2,4,8,7,5,3,7,7,1,3,8,8,
%T A282545 4,6,7,6,2,4,9,6,3,2,9,6,1,3,6,10,6,8,9,11,5,4,9,6,6,3,8,8,3,2,8,10,6,
%U A282545 9,11,9,1,5,9,9,6,2,7,6,1,4
%N A282545 Number of ways to write n as x^4 + y^2 + z^2 + w^2 with y^2 + 64*z^2 + 1024*y*z a square, where x,y,z,w are nonnegative integers with y > 0.
%C A282545 Conjecture: (i) a(n) > 0 for all n > 0.
%C A282545 (ii) Any nonnegative integer can be written as x^4 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that a*y^2 + b*y*z + c*z^2 is a square, whenever (a,b,c) is among the ordered triples (1,484,44), (1,666,9), (16,1336,169), (25,900,36).
%C A282545 (iii) For each c = 1, 49, any nonnegative integer can be written as x^4 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that 120*(x^2+y)*z + c*z^2 is a square.
%C A282545 By the linked JNT paper, each n = 0,1,2,... can be expressed as the sum of a fourth power and three squares.
%C A282545 See also A282463, A282494 and A282495 for similar conjectures.
%H A282545 Zhi-Wei Sun, <a href="/A282545/b282545.txt">Table of n, a(n) for n = 1..10000</a>
%H A282545 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 A282545 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017.
%e A282545 a(1) = 1 since 1 = 0^4 + 1^2 + 0^2 + 0^2 with 1^2 + 64*0^2 + 1024*1*0 = 1^2.
%e A282545 a(31) = 1 since 31 = 1^4 + 2^2 + 1^2 + 5^2 with 2^2 + 64*1^2 + 1024*2*1 = 46^2.
%e A282545 a(47) = 1 since 47 = 1^4 + 6^2 + 3^2 + 1^2 with 6^2 + 64*3^2 + 1024*6*3 = 138^2.
%e A282545 a(79) = 1 since 79 = 1^4 + 7^2 + 2^2 + 5^2 with 7^2 + 64*2^2 + 1024*7*2 = 121^2.
%e A282545 a(156) = 1 since 156 = 3^4 + 5^2 + 5^2 + 5^2 with 5^2 + 64*5^2 + 1024*5*5 = 165^2.
%e A282545 a(184) = 1 since 184 = 0^4 + 12^2 + 6^2 + 2^2 with 12^2 + 64*6^2 + 1024*12*6 = 276^2.
%e A282545 a(316) = 1 since 316 = 2^4 + 10^2 + 10^2 + 10^2 with 10^2 + 64*10^2 + 1024*10*10 = 330^2.
%e A282545 a(380) = 1 since 380 = 1^4 + 3^2 + 3^2 + 19^2 with 3^2 + 64*3^2 + 1024*3*3 = 99^2.
%e A282545 a(2383) = 1 since 2383 = 3^4 + 22^2 + 33^2 + 27^2 with 22^2 + 64*33^2 + 1024*22*33 = 902^2.
%t A282545 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A282545 Do[r=0;Do[If[SQ[n-x^4-y^2-z^2]&&SQ[y^2+1024y*z+64z^2],r=r+1],{x,0,(n-1)^(1/4)},{y,1,Sqrt[n-x^4]},{z,0,Sqrt[n-x^4-y^2]}];Print[n," ",r];Continue,{n,1,80}]
%Y A282545 Cf. A000118, A000290, A000583, A270969, A271518, A281976, A282463, A282494, A282495, A282542.
%K A282545 nonn
%O A282545 1,2
%A A282545 _Zhi-Wei Sun_, Feb 18 2017