OFFSET
1,3
COMMENTS
Conjecture: a(n) > 0 for all n > 0.
We have verified a(n) > 0 for all n = 1..10^6.
See also A343387 for a similar conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Natural numbers represented by [x^2/a] + [y^2/b] + [z^2/c], arXiv:1504.01608 [math.NT], 2015.
EXAMPLE
a(1) = 1 with 1 = 1^2 + [1^2/4] + [1^4/6].
a(2) = 1 with 2 = 1^2 + [2^2/4] + [1^4/6].
a(14) = 1 with 14 = 1^2 + [1^2/4] + [3^4/6].
a(32) = 1 with 32 = 4^2 + [8^2/4] + [1^4/6].
a(35) = 1 with 35 = 4^2 + [5^2/4] + [3^4/6].
a(77) = 1 with 77 = 8^2 + [1^2/4] + [3^4/6].
a(840) = 1 with 840 = 28^2 + [15^2/4] + [1^4/6].
MATHEMATICA
SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[n-Floor[x^2/4]-Floor[y^4/6]], r=r+1], {x, 1, Sqrt[4n+3]}, {y, 1, (6(n-Floor[x^2/4])+5)^(1/4)}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 13 2021
STATUS
approved