A349943 Number of ways to write n as a^4 + (b^4 + c^2 + d^2)/9, where a,b,c,d are nonnegative integers with c <= d.

1, 3, 5, 4, 3, 4, 3, 1, 1, 6, 9, 6, 2, 4, 7, 3, 3, 7, 9, 7, 7, 5, 4, 2, 3, 10, 11, 8, 2, 10, 10, 1, 5, 9, 15, 14, 6, 5, 5, 1, 4, 9, 12, 8, 2, 11, 7, 1, 4, 11, 21, 8, 6, 9, 8, 3, 3, 7, 9, 9, 4, 11, 9, 2, 3, 13, 14, 7, 7, 10, 10, 4, 3, 10, 18, 16, 3, 10, 7, 1, 4, 10, 15, 12, 11, 12, 11, 3, 3, 16, 29, 17, 5, 6, 14, 10, 3, 10, 18, 15, 14

Offset: 0

Author

Zhi-Wei Sun, Dec 05 2021

Keywords

• nonn

Comments

Conjecture 1: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 16^k*m (k = 0,1,2,... and m = 7, 8, 31, 39, 47, 79, 519).

This implies that each nonnegative rational number can be written as x^4 + 9*y^4 + z^2 + w^2 with x,y,z,w rational numbers.

Conjecture 2: Each n = 0,1,2,... can be written as a^4 + (4*b^4 + c^2 + d^2)/81 with a,b,c,d nonnegative integers.

This implies that each nonnegative rational number can be written as x^4 + 4*y^4 + z^2 + w^2 with x,y,z,w rational numbers.

We have verified Conjectures 1 and 2 for n <= 10^5.

It seems that each n = 0,1,2,... can be written as a^4 + (b^4 + c^2 + d^2)/m^2 with a,b,c,d nonnegative integers, provided that m is among the odd numbers 7, 11, 15, 17, 19, 21, ....

See also A349942 for a similar conjecture.

Examples

			a(7) = 1 with 7 = 1^4 + (1^4 + 2^2 + 7^2)/9.
a(8) = 1 with 8 = 0^4 + (0^4 + 6^2 + 6^2)/9.
a(31) = 1 with 31 = 1^4 + (1^4 + 10^2 + 13^2)/9.
a(39) = 1 with 39 = 1^4 + (3^4 + 6^2 + 15^2)/9.
a(47) = 1 with 47 = 1^4 + (3^4 + 3^2 + 18^2)/9.
a(79) = 1 with 79 = 1^4 + (1^4 + 5^2 + 26^2)/9.
a(519) = 1 with 519 = 1^4 + (3^4 + 15^2 + 66^2)/9.
		

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.

Programs

Mathematica

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[SQ[9(n-x^4)-y^4-z^2],r=r+1],{x,0,n^(1/4)},{y,0,(9(n-x^4))^(1/4)},{z,0,Sqrt[(9(n-x^4)-y^4)/2]}];tab=Append[tab,r],{n,0,100}];Print[tab]