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Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x-y+z+w a nonnegative cube, where x,y,z,w are integers with x >= y >= 0 and x >= |z| <= |w|.
8

%I #13 May 25 2016 22:02:06

%S 1,2,2,3,2,2,3,3,2,2,3,2,1,5,4,3,2,1,4,3,3,6,3,2,5,3,9,3,1,1,7,5,3,7,

%T 10,4,6,2,10,2,6,2,12,7,2,5,9,3,3,6,13,3,8,3,18,3,8,5,7,3,3,5,13,8,5,

%U 3,19,4,7,7,16,1,11,5,14,7,2,3,12,5,4

%N Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x-y+z+w a nonnegative cube, where x,y,z,w are integers with x >= y >= 0 and x >= |z| <= |w|.

%C Conjecture: a(n) > 0 for all n = 0,1,2,....

%C In the latest version of arXiv:1605.03074, the authors showed that any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w integers such that x + y + z + w is a cube (or a square).

%C For more conjectural refinements of Lagrange's four-square theorem, see the author's preprint arXiv:1604.06723.

%H Zhi-Wei Sun, <a href="/A273458/b273458.txt">Table of n, a(n) for n = 0..10000</a>

%H Yu-Chen Sun and Zhi-Wei Sun, <a href="http://arxiv.org/abs/1605.03074">Two refinements of Lagrange's four-square theorem</a>, arXiv:1605.03074 [math.NT], 2016.

%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(12) = 1 since 12 = 3^2 + 1^2 + (-1)^2 + (-1)^2 with 3 - 1 + (-1) + (-1) = 0^3.

%e a(17) = 1 since 17 = 2^2 + 0^2 + 2^2 + (-3)^2 with 2 - 0 + 2 + (-3) = 1^3.

%e a(28) = 1 since 28 = 3^2 + 1^2 + 3^2 + 3^2 with 3 - 1 + 3 + 3 = 2^3.

%e a(29) = 1 since 29 = 3^2 + 0^2 + 2^2 + (-4)^2 with 3 - 0 + 2 + (-4) = 1^3.

%e a(71) = 1 since 71 = 5^2 + 1^2 + 3^2 + (-6)^2 with 5 - 1 + 3 + (-6) = 1^3.

%e a(149) = 1 since 149 = 8^2 + 0^2 + 2^2 + (-9)^2 with 8 - 0 + 2 + (-9) = 1^3.

%e a(188) = 1 since 188 = 13^2 + 3^2 + 1^2 + (-3)^2 with 13 - 3 + 1 + (-3) = 2^3.

%e a(284) = 1 since 284 = 15^2 + 5^2 + 3^2 + (-5)^2 with 15 - 5 + 3 + (-5) = 2^3.

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]

%t CQ[n_]:=CQ[n]=n>=0&&IntegerQ[n^(1/3)]

%t Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&CQ[x-y+(-1)^j*z+(-1)^k*Sqrt[n-x^2-y^2-z^2]],r=r+1],{y,0,(n/2)^(1/2)},{x,y,Sqrt[n-y^2]},{z,0,Min[x,Sqrt[(n-x^2-y^2)/2]]},{j,0,Min[1,z]},{k,0,Min[1,Sqrt[n-x^2-y^2-z^2]]}];

%t Print[n," ",r];Continue,{n,0,80}]

%Y Cf. A000118, A000290, A000578, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A270969, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134, A273278, A273294, A273302, A273404, A273429, A273432, A273568.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, May 22 2016