OFFSET
0,3
COMMENTS
3-4-5-6 Conjecture: a(n) > 0 for all n >= 0.
We have verified a(n) > 0 for all n = 0..10^6.
Conjecture verified up to 2*10^9. - Giovanni Resta, Apr 28 2021
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, Natural numbers represented by floor(x^2/a) + floor(y^2/b) + floor(z^2/c), arXiv:1504.01608 [math.NT], 2015.
EXAMPLE
a(0) = 1 with 0 = [1^3/3] + [1^3/4] + [1^3/5] + [1^6/6].
a(1) = 1 with 1 = [1^3/3] + [1^3/4] + [2^3/5] + [1^6/6].
a(4) = 1 with 4 = [2^3/3] + [2^3/4] + [1^3/5] + [1^6/6].
a(6) = 1 with 6 = [1^3/3] + [3^3/4] + [1^3/5] + [1^6/6].
a(8) = 1 with 8 = [2^3/3] + [3^3/4] + [1^3/5] + [1^6/6].
a(60) = 1 with 60 = [3^3/3] + [4^3/4] + [5^3/5] + [2^6/6].
a(81) = 1 with 81 = [2^3/3] + [6^3/4] + [5^3/5] + [1^6/6].
a(300) = 1 with 300 = [7^3/3] + [5^3/4] + [9^3/5] + [2^6/6].
a(4434) = 1 with 4434 = [11^3/3] + [4^3/4] + [19^3/5] + [5^6/6].
MATHEMATICA
CQ[n_]:=CQ[n]=n>0&&IntegerQ[n^(1/3)];
tab={}; Do[r=0; Do[If[CQ[3(n-Floor[x^6/6]-Floor[y^3/5]-Floor[z^3/4])+s], r=r+1], {s, 0, 2}, {x, 1, (6n+5)^(1/6)}, {y, 1, (5(n-Floor[x^6/6])+4)^(1/3)}, {z, 1, (4(n-Floor[x^6/6]-Floor[y^3/5])+3)^(1/3)}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 13 2021
STATUS
approved