A282542 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that x + 3*y + 5*z and (at least) one of y,z,w are squares.

1, 2, 2, 2, 2, 1, 1, 1, 1, 3, 3, 2, 1, 2, 4, 2, 2, 4, 5, 3, 2, 2, 2, 2, 1, 5, 5, 2, 1, 5, 8, 1, 2, 3, 3, 3, 2, 3, 5, 5, 2, 8, 5, 1, 1, 6, 6, 1, 2, 5, 9, 5, 4, 2, 5, 5, 2, 5, 4, 5, 2, 1, 5, 3, 2, 7, 9, 5, 2, 3, 6, 2, 2, 8, 9, 5, 3, 5, 9, 2, 1

Offset: 0

Author

Zhi-Wei Sun, Feb 17 2017

Keywords

• nonn

Comments

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

This is stronger than the 1-3-5 conjecture (cf. A271518).

By the linked JNT paper, any nonnegative integer can be expressed as the sum of a fourth power and three squares.

Examples

			a(12) = 1 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1 + 3*1 + 5*1 = 3^2 and 1 = 1^2.
a(28) = 1 since 28 = 1^2 + 1^2 + 1^2 + 5^2 with 1 + 3*1 + 5*1 = 3^2 and 1 = 1^2.
a(47) = 1 since 47 = 3^2 + 1^2 + 6^2 + 1^2 with 3 + 3*1 + 5*6 = 6^2 and 1 = 1^2.
a(92) = 1 since 92 = 1^2 + 1^2 + 9^2 + 3^2 with 1 + 3*1 + 5*1 = 3^2 and 9 = 3^2.
a(188) = 1 since 188 = 7^2 + 9^2 + 3^2 + 7^2 with 7 + 3*9 + 5*3 = 7^2 and 9 = 3^2.
a(248) = 1 since 248 = 10^2 + 2^2 + 0^2 + 12^2 with 10 + 3*2 + 5*0 = 4^2 and 0 = 0^2.
a(388) = 1 since 388 = 13^2 + 1^2 + 13^2 + 7^2 with 13 + 3*1 + 5*13 = 9^2 and 1 = 1^2.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
• Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
• Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017.

Crossrefs

Cf. A000118, A000290, A270969, A271518, A281976.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&(SQ[y]||SQ[z]||SQ[Sqrt[n-x^2-y^2-z^2]])&&SQ[x+3y+5z],r=r+1],{x,0,n^(1/2)},{y,0,Sqrt[n-x^2]},{z,0,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,0,80}]