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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 06 2021
STATUS
approved