A282014 Number of ways to write n as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that both x and 121*x + 48*(y-z) are squares.

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

Offset

0,2

Comments

Conjecture: a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 8, 29, 31, 40, 94, 104, 143, 319, 671).

The author has proved that any nonnegative integer can be written as the sum of a fourth power and three squares.

We have verified a(n) > 0 for all n = 0..10^7.

See also A281976, A281977 and A282013 for similar conjectures.

Qing-Hu Hou at Tianjin University verified a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, Jun 02 2019

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(8) = 1 since 8 = 0^2 + 2^2 + 2^2 + 0^2 with 0 = 0^2 and 121*0 + 48*(2-2) = 0^2.
a(29) = 1 since 29 = 0^2 + 5^2 + 2^2 + 0^2 with 0 = 0^2 and 121*0 + 48*(5-2) = 12^2.
a(31) = 1 since 31 = 1^2 + 2^2 + 1^2 + 5^2 with 1 = 1^2 and 121*1 + 48*(2-1) = 13^2.
a(40) = 1 since 40 = 4^2 + 2^2 + 2^2 + 4^2 with 4 = 2^2 and 121*4 + 48*(2-2) = 22^2.
a(94) = 1 since 94 = 0^2 + 6^2 + 3^2 + 7^2 with 0 = 0^2 and 121*0 + 48*(6-3) = 12^2.
a(104) = 1 since 104 = 4^2 + 6^2 + 6^2 + 4^2 with 4 = 2^2 and 121*4 + 48*(6-6) = 22^2.
a(143) = 1 since 143 = 1^2 + 6^2 + 5^2 + 9^2 with 1 = 1^2 and 121*1 + 48*(6-5) = 13^2.
a(319) = 1 since 319 = 1^2 + 17^2 + 2^2 + 5^2 with 1 = 1^2 and 121*1 + 48*(17-2) = 29^2.
a(671) = 1 since 671 = 9^2 + 5^2 + 23^2 + 6^2 with 9 = 3^2 and 121*9 + 48*(5-23) = 15^2.

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
Do[r=0; Do[If[SQ[n-x^4-y^2-z^2]&&SQ[121x^2+48(y-z)], r=r+1], {x, 0, n^(1/4)}, {y, 0, Sqrt[n-x^4]}, {z, 0, Sqrt[n-x^4-y^2]}]; Print[n, " ", r]; Continue, {n, 0, 80}]

Crossrefs

Cf. A000118, A000290, A270969, A271518, A281939, A281941, A281975, A281976, A281977, A282013.

Sequence in context: A334236 A030423 A130631 * A241539 A213512 A218774

Adjacent sequences: A282011 A282012 A282013 * A282015 A282016 A282017

Keyword

nonn

Author

Zhi-Wei Sun, Feb 04 2017

Status

approved