OFFSET
1,5
COMMENTS
Conjecture: a(n) > 0 for all n > 1.
We have verified a(n) > 0 for all 1 < n <= 3*10^5.
Conjecture verified up to 10^10. - Giovanni Resta, Apr 14 2021
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(2) = 1 with 2 = 1^3 + [1^3/2] + [1^3/3] + 2^0.
a(3) = 1 with 3 = 1^3 + [1^3/2] + [1^3/3] + 2^1.
a(4) = 1 with 4 = 1^3 + [1^3/2] + [2^3/3] + 2^0.
a(6) = 1 with 6 = 1^3 + [2^3/2] + [1^3/3] + 2^0.
a(8) = 1 with 8 = 1^3 + [2^3/2] + [2^3/3] + 2^0.
a(10) = 1 with 10 = 2^3 + [1^3/2] + [1^3/3] + 2^1.
a(103) = 1 with 103 = 3^3 + [1^3/2] + [6^3/3] + 2^2.
MATHEMATICA
CQ[n_]:=CQ[n]=n>0&&IntegerQ[n^(1/3)];
tab={}; Do[r=0; Do[If[CQ[n-Floor[x^3/2]-Floor[y^3/3]-2^z], r=r+1], {x, 1, (2n-1)^(1/3)}, {y, 1, (3(n-Floor[x^3/2])-1)^(1/3)}, {z, 0, Log[2, n-Floor[x^3/2]-Floor[y^3/3]]}]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 14 2021
STATUS
approved