cp's OEIS Frontend

A273458 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|.

%I A273458 #13 May 25 2016 22:02:06
%S A273458 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 A273458 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 A273458 3,19,4,7,7,16,1,11,5,14,7,2,3,12,5,4
%N A273458 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 A273458 Conjecture: a(n) > 0 for all n = 0,1,2,....
%C A273458 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 A273458 For more conjectural refinements of Lagrange's four-square theorem, see the author's preprint arXiv:1604.06723.
%H A273458 Zhi-Wei Sun, <a href="/A273458/b273458.txt">Table of n, a(n) for n = 0..10000</a>
%H A273458 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 A273458 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 A273458 a(12) = 1 since 12 = 3^2 + 1^2 + (-1)^2 + (-1)^2 with 3 - 1 + (-1) + (-1) = 0^3.
%e A273458 a(17) = 1 since 17 = 2^2 + 0^2 + 2^2 + (-3)^2 with 2 - 0 + 2 + (-3) = 1^3.
%e A273458 a(28) = 1 since 28 = 3^2 + 1^2 + 3^2 + 3^2 with 3 - 1 + 3 + 3 = 2^3.
%e A273458 a(29) = 1 since 29 = 3^2 + 0^2 + 2^2 + (-4)^2 with 3 - 0 + 2 + (-4) = 1^3.
%e A273458 a(71) = 1 since 71 = 5^2 + 1^2 + 3^2 + (-6)^2 with 5 - 1 + 3 + (-6) = 1^3.
%e A273458 a(149) = 1 since 149 = 8^2 + 0^2 + 2^2 + (-9)^2 with 8 - 0 + 2 + (-9) = 1^3.
%e A273458 a(188) = 1 since 188 = 13^2 + 3^2 + 1^2 + (-3)^2 with 13 - 3 + 1 + (-3) = 2^3.
%e A273458 a(284) = 1 since 284 = 15^2 + 5^2 + 3^2 + (-5)^2 with 15 - 5 + 3 + (-5) = 2^3.
%t A273458 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A273458 CQ[n_]:=CQ[n]=n>=0&&IntegerQ[n^(1/3)]
%t A273458 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 A273458 Print[n," ",r];Continue,{n,0,80}]
%Y A273458 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 A273458 nonn
%O A273458 0,2
%A A273458 _Zhi-Wei Sun_, May 22 2016