A349956 Number of ways to write n as x^2 + 2*y^2 + (z^4 + 4*w^4)/5 with x,y,z,w nonnegative integers.

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

Offset

0,2

Comments

a(n) > 0 for all n <= 10^5.

Conjecture: If m is 5 or 65 or 85, then each n = 0,1,2,... can be written as x^2 + 2*y^2 + (z^4 + 4*w^4)/m with x,y,z,w nonnegative integers.

It seems that there are infinitely many positive squarefree numbers m (including 3, 5, 15, 23, 31, 33, 37, 55, 59, 67, 69, 71, 89, 93, 97, 111, 113, 115) such that every n = 0,1,2,... can be written as x^4 + 2*y^4 + (z^2 + 11*w^2)/m with x,y,z,w nonnegative integers.

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, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.

Example

a(11) = 1 with 11 = 3^2 + 2*1^2 + (0^4 + 4*0^4)/5.
a(14) = 1 with 14 = 1^2 + 2*0^2 + (1^4 + 4*2^4)/5.
a(78) = 1 with 78 = 7^2 + 2*0^2 + (3^4 + 4*2^4)/5.
a(155) = 1 with 155 = 11^2 + 2*3^2 + (2^4 + 4*2^4)/5.
a(174) = 1 with 174 = 7^2 + 2*0^2 + (5^4 + 4*0^4)/5.

Mathematica

QQ[n_]:=QQ[n]=IntegerQ[n^(1/4)];
tab={}; Do[r=0; Do[If[QQ[5(n-2x^2-y^2)-4z^4], r=r+1], {x, 0, Sqrt[n/2]}, {y, 0, Sqrt[n-2x^2]}, {z, 0, (5(n-2x^2-y^2)/4)^(1/4)}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]

Crossrefs

Cf. A000290, A000583, A349942, A349945.

Sequence in context: A089367 A130192 A175064 * A104564 A160764 A156384

Adjacent sequences: A349953 A349954 A349955 * A349957 A349958 A349959

Keyword

nonn

Author

Zhi-Wei Sun, Dec 06 2021

Status

approved