OFFSET
1,2
COMMENTS
Conjecture: a(n) > 0 for all n > 0.
This has been verified for n up to 10^6.
As (x^2 + y^2)/2 = ((x+y)/2)^2 + ((x-y)/2)^2, the conjecture gives a new refinement of Lagrange's four-square theorem.
See also A350012 for a similar conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167--190.
Zhi-Wei Sun, Restricted sums of four squares, Int. J. Number Theory 15(2019), 1863-1893. See also arXiv:1701.05868 [math.NT].
Zhi-Wei Sun, New Conjectures in Number Theory and Combinatorics (in Chinese), Harbin Institute of Technology Press, 2021.
EXAMPLE
a(1) = 1 with 1 = 0^4 + 0^2 + (1^2 + 4^0)/2.
a(23) = 1 with 23 = 1^4 + 3^2 + (5^2 + 4^0)/2.
a(79) = 1 with 79 = 1^4 + 2^2 + (12^2 + 4^1)/2.
a(1199) = 1 with 1199 = 5^4 + 18^2 + (22^2 + 4^2)/2.
a(3679) = 1 with 3679 = 5^4 + 2^2 + (78^2 + 4^2)/2.
a(6079) = 1 with 6079 = 3^4 + 42^2 + (92^2 + 4^1)/2.
a(33439) = 1 with 33439 = 1^4 + 175^2 + (75^2 + 4^0)/2.
a(50399) = 1 with 50399 = 13^4 + 135^2 + (85^2 + 4^0)/2.
a(207439) = 1 with 207439 = 1^4 + 142^2 + (612^2 + 4^1)/2.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[2(n-x^4-y^2)-4^z], r=r+1], {x, 0, (n-1)^(1/4)}, {y, 0, Sqrt[n-1-x^4]}, {z, 0, Log[4, 2(n-x^4-y^2)]}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 08 2021
STATUS
approved