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

Author

Zhi-Wei Sun, Dec 06 2021

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000290, A000583, A349942, A349945.

Programs

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]