A281659 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with 9*x^2 + 16*y^2 + 24*z^2 + 48*w^2 a square, where x,y,z,w are nonnegative integers.

1, 2, 2, 2, 2, 1, 4, 1, 2, 3, 4, 1, 5, 1, 5, 1, 2, 2, 8, 1, 2, 5, 2, 2, 4, 3, 2, 8, 1, 5, 4, 2, 2, 3, 3, 4, 4, 3, 2, 2, 4, 2, 6, 1, 1, 3, 2, 3, 5, 2, 3, 7, 3, 1, 7, 1, 5, 2, 2, 4, 1, 1, 6, 7, 2, 1, 7, 2, 4, 5, 6, 0, 8, 3, 6, 8, 5, 3, 4, 2, 2

Offset: 0

Author

Zhi-Wei Sun, Jan 26 2017

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 except for n = 71, 85.

(ii) Any nonnegative integer n not among 39, 71 and 649 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 16*y^2 + 24*z^2 + 32*w^2 a square, where x,y,z,w are integers.

(iii) Each nonnegative integer n not among 5, 7, 23, 47, 93, 103, 109, 151, 191 and 911 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 3*y^2 + 8*z^2 + 16*w^2 a square, where x,y,z,w are integers.

(iv) Any nonnegative integer n not among 15, 19, 71, 103, 191 and 559 can be written as x^2 + y^2 + z^2 + w^2 with 9*x^2 + 48*y^2 + 64*z^2 + 96*w^2 a square, where x,y,z,w are integers.

(v) Any nonnegative integer n not among 5, 93, 95, 161 and 309 can be written as x^2 + y^2 + z^2 + w^2 with 16*x^2 + 25*y^2 + 48*z^2 + 128*w^2 a square, where x,y,z,w are integers.

(vi) Each nonnegative integer not among 13, 19, 39, 41, 71, 109, 131, 193, 233, 377, 415 and 941 can be written as x^2 + y^2 + z^2 + w^2 with x^2 + 24*y^2 + 48*z^2 + 144*w^2 a square, where x,y,z,w are integers.

We have verified part (i) of the conjecture for n up to 1.2*10^5.

Examples

			a(11) = 1 since 11 = 3^2 + 1^2 + 1^2 + 0^2 with 9*3^2 + 16*1^2 + 24*1^2 + 48*0 = 11^2.
a(170) = 1 since 170 = 3^2 + 6^2 + 2^2 + 11^2 with 9*3^2 + 16*6^2 + 24*2^2 + 48*11^2 = 81^2.
a(305) = 1 since 305 = 0^2 + 15^2 + 4^2 + 8^2 with 9*0^2 + 16*15^2 + 24*4^2 + 48*8^2 = 84^2.
		

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, A271510, A271513, A271518.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[9x^2+16y^2+24z^2+48*(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];Label[aa];Continue,{n,0,80}]