A271644 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that w*x + 2*x*y + 2*y*z is a square, where w is a positive integer and x,y,z are nonnegative integers.

1, 3, 1, 1, 4, 4, 1, 3, 4, 4, 2, 2, 5, 2, 1, 1, 8, 8, 2, 5, 7, 3, 2, 4, 8, 7, 3, 2, 6, 4, 4, 3, 7, 6, 2, 4, 6, 4, 3, 4, 9, 4, 3, 4, 8, 4, 1, 2, 5, 7, 4, 7, 10, 11, 3, 2, 5, 5, 2, 2, 7, 4, 2, 1, 8, 9, 2, 8, 14, 9, 1, 8, 8, 6, 5, 4, 8, 2, 3, 5

Offset: 1

Author

Zhi-Wei Sun, Apr 11 2016

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 3, 7, 15, 47, 71, 379, 4^k (k = 0,1,2,...).

(ii) If a, b and c are positive integers with a <= b <= c, gcd(a,b,c) squarefree, and the triple (a,b,c) not equal to (1,2,2), then not all natural numbers can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers and a*w*x + b*x*y + c*y*z a square.

(iii) Let a,b,c be positive integers with gcd(a,b,c) squarefree. Then every n = 0,1,2,... can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers such that a*x*y + b*y*z + c*z*x is a square, if and only if {a,b,c} is among {1,2,3}, {1,3,8}, {1,8,13}, {2,4,45}, {4,5,7}, {4,7,23}, {5,8,9}, {11,16,31}.

Clearly, part (i) of this conjecture is stronger than Lagrange's four-square theorem.

See also A271510, A271513, A271518 and A271608 for related conjectures.

Examples

			a(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1*0 + 2*0*0 + 2*0*0 = 0^2.
a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with 1*1 + 2*1*0 + 2*0*1 = 1^2.
a(7) = 1 since 7 = 1^2 + 1^2 + 2^2 + 1^2 with 1*1 + 2*1*2 + 2*2*1 = 3^2.
a(11) = 2 since 11 = 1^2 + 1^2 + 0^2 + 3^2 with 1*1 + 2*1*0 + 2*0*3 = 1^2, and 11 = 1^2 + 3^2 + 1^2 + 0^2 with 1*3 + 2*3*1 + 2*1*0 = 3^2.
a(12) = 2 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1*1 + 2*1*1 + 2*1*3 = 3^2, and 12 = 2^2 + 2^2 + 0^2 + 2^2  with 2*2 + 2*2*0 + 2*0*2 = 2^2.
a(15) = 1 since 15 = 3^2 + 1^2 + 1^2 + 2^2 with 3*1 + 2*1*1 + 2*1*2 = 3^2.
a(47) = 1 since 47 = 1^2 + 1^2 + 6^2 + 3^2 with 1*1 + 2*1*6 + 2*6*3 = 7^2.
a(71) = 1 since 71 = 3^2 + 3^2 + 2^2 + 7^2 with 3*3 + 2*3*2 + 2*2*7 = 7^2.
a(379) = 1 since 379 = 3^2 + 3^2 + 0^2 + 19^2 with 3*3 + 2*3*0 + 2*0*19 = 3^2.
		

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 [math.GM], 2016.

Crossrefs

Cf. A000118, A000290, A271510, A271513, A271518, A271608.

Programs

Mathematica

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