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%I #37 Mar 14 2017 20:45:47
%S 1,1,1,1,2,2,3,1,2,3,1,2,3,3,2,2,3,3,2,2,4,3,5,2,1,3,2,4,2,3,4,2,2,3,
%T 1,3,5,3,4,1,3,4,2,2,5,2,1,3,2,3,1,4,3,2,6,3,1,4,3,3,2,4,6,1,2,6,3,3,
%U 6,3,6,2,5,3,1,6,7,5,2,4,5
%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 72*x^3 + (y-z)^3 is an even square.
%C Conjecture: (i) a(n) > 0 for any nonnegative integer n. Also, each n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^3 + ((y-z)/2)^3 is a square (or twice a square).
%C (ii) Let a and b be positive integers with gcd(a,b) squarefree. Then, every n = 0,1,2,... can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that a*x^3 + b*(y-z)^3 is a square, if and only if (a,b) is among the ordered pairs (1,1), (1,9), (2,18), (8,1), (9,5), (9,8), (9,40), (16,2), (18,16), (25,16), (72,1).
%C We have verified that a(n) > 0 for all n = 0..2*10^6.
%H Zhi-Wei Sun, <a href="/A282863/b282863.txt">Table of n, a(n) for n = 0..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(56) = 1 since 56 = 0^2 + 6^2 + 2^2 + 4^2 with 72*0^3 + (6-2)^3 = 8^2.
%e a(120) = 1 since 120 = 4^2 + 2^2 + 10^2 + 0^2 with 72*4^3 + (2-10)^3 = 64^2.
%e a(159) = 1 since 159 = 2^2 + 3^2 + 11^2 + 5^2 with 72*2^3 + (3-11)^3 = 8^2.
%e a(1646) = 1 since 1646 = 5^2 + 10^2 + 0^2 + 39^2 with 72*5^3 + (10-0)^3 = 100^2.
%e a(1784) = 1 since 1784 = 12^2 + 22^2 + 30^2 + 16^2 with 72*12^3 + (22-30)^3 = 352^2.
%e a(3914) = 1 since 3914 = 2^2 + 45^2 + 21^2 + 38^2 with 72*2^3 + (45-21)^3 = 120^2.
%e a(5864) = 1 since 5864 = 50^2 + 0^2 + 0^2 + 58^2 with 72*50^3 + (0-0)^3 = 3000^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[72x^3+(y-z)^3]&&Mod[y-z,2]==0,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,0,80}]
%Y Cf. A000118, A000290, A000578, A271518, A281976, A283617.
%K nonn
%O 0,5
%A _Zhi-Wei Sun_, Mar 14 2017
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