OFFSET
1,2
COMMENTS
Conjecture: For any positive integer n, the set {x^3 + 2*y^3 + 3*z^3: x,y,z = 0,...,n-1 and x*y*z is a cube} contains a complete system of residues modulo n.
See also A273488 for a similar conjecture.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..300
EXAMPLE
a(2) = 4 since 0^3 + 2*0^3 + 3*1^3 == 7 (mod 2) with 0*0*1 = 0^3, 0^3 + 2*1^3 + 3*1^3 == 7 (mod 2) with 0*1*1 = 0^3, 1^3 + 2*0^3 + 3*0^3 == 7 (mod 2) with 1*0*0 = 0^3, and 1^3 + 2*1^3 + 3*0^3 == 7 (mod 2) with 1*1*0 = 0^3.
a(7) = 1 since 0^3 + 2*0^3 + 3*0^3 == 7 (mod 7) with 0*0*0 = 0^3.
MATHEMATICA
CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
Do[r=0; Do[If[Mod[x^3+2y^3+3z^3-7, n]==0&&CQ[x*y*z], r=r+1], {x, 0, n-1}, {y, 0, n-1}, {z, 0, n-1}]; Print[n, " ", r]; Continue, {n, 1, 60}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 28 2016
STATUS
approved