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

0,2

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.

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.

Example

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.

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}]

Crossrefs

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

Sequence in context: A231168 A276830 A064741 * A345202 A308619 A114294

Adjacent sequences: A281656 A281657 A281658 * A281660 A281661 A281662

Keyword

nonn

Author

Zhi-Wei Sun, Jan 26 2017

Status

approved