OFFSET
0,2
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.
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(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.
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]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 05 2021
STATUS
approved