A262357 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with w^2*x^2 + 5*x^2*y^2 + 80*y^2*z^2 + 20*z^2*w^2 a square, where w is a positive integer and x,y,z are nonnegative integers.

1, 2, 1, 1, 4, 3, 2, 2, 4, 5, 1, 1, 6, 3, 2, 1, 6, 7, 2, 4, 8, 6, 2, 3, 8, 9, 3, 2, 8, 5, 2, 2, 6, 6, 2, 4, 9, 5, 4, 5, 8, 5, 1, 1, 10, 5, 3, 1, 5, 9, 3, 6, 10, 10, 6, 3, 5, 5, 2, 2, 12, 3, 5, 1, 13, 9, 3, 6, 10, 9

Offset: 1

Author

Zhi-Wei Sun, Apr 17 2016

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*m (k = 0,1,2,... and m = 1, 3, 11, 43, 547, 763, 1739, 6783).

(ii) For each quadruples (a,b,c,d) = (1,3,78,27), (1,3,222,75), (4,12,81,108), (6,27,25,75), (7,21,112,32), any positive integer can be written as w^2 + x^2 + y^2 + z^2 with a*w^2*x^2 + b*x^2*y^2 + c*y^2*z^2 + d*z^2*w^2 a square, where w is a positive integer and x,y,z are integers.

(iii) Each n = 0,1,2,.... can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z integers such that w^2*x^2 + 4*x^2*y^2 + 44*y^2*z^2 + 16*z^2*w^2 = 5*t^2 for some integer t.

See also A268507, A269400, A271510, A271513, A271518, A271665, A271714, A271721, A271724, A271775, A271778 and A271824 for other conjectures refining Lagrange's four-square theorem.

Examples

			a(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1 > 0 and 1^2*0^2 + 5*0^2*0^2 + 80*0^2*0^2 + 20*0^2*1^2 = 0^2.
a(2) = 2 since 2 = 1^2 + 0^2 + 1^2 + 0^2 with 1 > 0 and 1^2*0^2 + 5*0^2*1^2 + 80*1^2*0^2 + 20*0^2*1^2 = 0^2, and also 2 = 1^2 + 1^2 + 0^2 + 0^2 with 1 > 0 and 1^2*1^2 + 5*1^2*0^2 + 80*0^2*0^2 + 20*0^2*1^2 = 1^2.
a(3) = 1 since 3 = 1^2 + 0^2 + 1^2 + 1^2 with 1 > 0 and 1^2*0^2 + 5*0^2*1^2 + 80*1^2*1^2 + 20*1^2*1^2 = 10^2.
a(11) = 1 since 11 = 1^2 + 0^2 + 1^2 + 3^2 with 1 > 0 and
1^2*0^2 + 5*0^2*1^2 + 80*1^2*3^2 + 20*3^2*1^2 = 30^2.
a(43) = 1 since 43 = 3^2 + 0^2 + 3^2 + 5^2 with 3 > 0 and 3*0^2 + 5*0^2*3^2 + 80*3^2*5^2 + 20*5^2*3^2 = 150^2.
a(547) = 1 since 547 = 3^2 + 0^2 + 3^2 + 23^2 with 3 > 0 and 3^2*0^2 + 5*0^2*3^2 + 80*3^2*23^2 + 20*23^2*3^2 = 690^2.
a(763) = 1 since 763 = 13^2 + 20^2 + 13^2 + 5^2 with 13 > 0 and 13^2*20^2 + 5*20^2*13^2 + 80*13^2*5^2 + 20*5^2*13^2 = 910^2.
a(1739) = 1 since 1739 = 15^2 + 16^2 + 27^2 + 23^2 with 15 > 0 and 15^2*16^2 + 5*16^2*27^2 + 80*27^2*23^2 + 20*23^2*15^2 = 5850^2.
a(6783) = 1 since 6783 = 17^2 + 73^2 + 18^2 + 29^2 with 17 > 0 and 17^2*73^2 + 5*73^2*18^2 + 80*18^2*29^2 + 20*29^2*17^2 = 6069^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, arxiv:1604.06723, 2016.

Crossrefs

Cf. A000118, A000290, A268507, A269400, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(n-x^2-y^2-z^2)*x^2+5*x^2*y^2+80*y^2*z^2+20*z^2*(n-x^2-y^2-z^2)],r=r+1],{x,0,Sqrt[n-1]},{y,0,Sqrt[n-1-x^2]},{z,0,Sqrt[n-1-x^2-y^2]}];Print[n," ",r];Continue,{n,1,70}]