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A273287
Number of triples (x,y,z) with x,y,z in the set {0,...,n-1} such that x*y*z is a cube and x^3 + 2*y^3 + 3*z^3 == 7 (mod n).
2
1, 4, 7, 10, 14, 18, 1, 22, 27, 29, 34, 37, 35, 7, 47, 46, 54, 56, 44, 60, 17, 71, 71, 72, 76, 60, 80, 15, 92, 93, 81, 98, 104, 109, 25, 115, 118, 90, 91, 121, 129, 33, 114, 142, 135, 142, 145, 147, 22, 153, 163, 121, 168, 176, 174, 29, 135, 182, 185, 184
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
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
Sequence in context: A310690 A310691 A161187 * A310692 A300306 A310693
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Aug 28 2016
STATUS
approved