cp's OEIS Frontend

A273278 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^2*y^2 + 3*y^2*z^2 + 2*z^2*w^2 is a square.

%I A273278 #7 May 19 2016 11:27:45
%S A273278 1,4,4,2,4,8,4,1,4,8,8,4,2,9,6,3,4,12,9,6,8,8,5,2,4,14,15,6,1,14,9,2,
%T A273278 4,9,12,8,8,9,11,1,8,18,7,4,4,17,8,3,2,12,18,9,9,17,15,4,6,8,8,10,3,
%U A273278 15,13,5,4,22,15,6,12,15,13
%N A273278 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x^2*y^2 + 3*y^2*z^2 + 2*z^2*w^2 is a square.
%C A273278 Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 4^k*m (k = 0,1,2,... and m = 7, 39, 87, 183, 231, 807, 879, 959, 1479, 2391, 2519, 2759, 4359, 10887).
%C A273278 See part (ii) of the conjecture in A269400 for similar conjectures.
%C A273278 For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.
%H A273278 Zhi-Wei Sun, <a href="/A273278/b273278.txt">Table of n, a(n) for n = 0..10000</a>
%H A273278 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 A273278 a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 2^2*1^2 + 3*1^2*1^2 + 2*1^2*1^2 = 3^2.
%e A273278 a(39) = 1 since 39 = 2^2 + 1^2 + 5^2 + 3^2 with 2^2*1^2 + 3*1^2*5^2 + 2*5^2*3^2 = 23^2.
%e A273278 a(87) = 1 since 87 = 2^2 + 1^2 + 1^2 + 9^2 with 2^2*1^2 + 3*1^2*1^2 + 2*1^2*9^2 = 13^2.
%e A273278 a(183) = 1 since 183 = 10^2 + 7^2 + 5^2 + 3^2 with 10^2*7^2 + 3*7^2*5^2 + 2*5^2*3^2 = 95^2.
%e A273278 a(231) = 1 since 231 = 10^2 + 1^2 + 9^2 + 7^2 with 10^2*1^2 + 3*1^2*9^2 + 2*9^2*7^2 = 91^2.
%e A273278 a(807) = 1 since 807 = 10^2 + 23^2 + 3^2 + 13^2 with 10^2*23^2 + 3*23^2*3^2 + 2*3^2*13^2 = 265^2.
%e A273278 a(879) = 1 since 879 = 14^2 + 11^2 + 21^2 + 11^2 with 14^2*11^2 + 3*11^2*21^2 + 2*21^2*11^2 = 539^2.
%e A273278 a(959) = 1 since 959 = 10^2 + 15^2 + 25^2 + 3^2 with 10^2*15^2 + 3*15^2*25^2 + 2*25^2*3^2 = 675^2.
%e A273278 a(1479) = 1 since 1479 = 34^2 + 11^2 + 11^2 + 9^2 with 34^2*11^2 + 3*11^2*11^2 + 2*11^2*9^2 = 451^2.
%e A273278 a(2391) = 1 since 2391 = 34^2 + 11^2 + 5^2 + 33^2 with 34^2*11^2 + 3*11^2*5^2 + 2*5^2*33^2 = 451^2.
%e A273278 a(2519) = 1 since 2519 = 42^2 + 1^2 + 27^2 + 5^2 with 42^2*1^2 + 3*1^2*27^2 + 2*27^2*5^2 = 201^2.
%e A273278 a(2759) = 1 since 2759 = 26^2 + 21^2 + 11^2 + 39^2 with 26^2*21^2 + 3*21^2*11^2 + 2*11^2*39^2 = 909^2.
%e A273278 a(4359) = 1 since 4359 = 46^2 + 19^2 + 19^2 + 39^2 with 46^2*19^2 + 3*19^2*19^2 + 2*19^2*39^2 = 1501^2.
%e A273278 a(10887) = 1 since 10887 = 31^2 + 85^2 + 51^2 + 10^2 with 31^2*85^2 + 3*85^2*51^2 + 2*51^2*10^2 = 7990^2.
%t A273278 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A273278 Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[x^2*y^2+3*y^2*z^2+2z^2*(n-x^2-y^2-z^2)],r=r+1],{x,0,Sqrt[n]},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,0,70}]
%Y A273278 Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273110, A273134.
%K A273278 nonn
%O A273278 0,2
%A A273278 _Zhi-Wei Sun_, May 18 2016