A273488 - OEIS

A273488

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 + 5*y^3 + 24*z^3 == 2 (mod n).

1, 2, 1, 1, 2, 5, 5, 3, 6, 6, 7, 2, 13, 8, 12, 10, 17, 2, 15, 9, 18, 24, 12, 8, 12, 23, 15, 11, 21, 25, 30, 14, 29, 27, 23, 25, 34, 29, 42, 19, 42, 35, 57, 16, 69, 45, 41, 23, 45, 43, 43, 34, 60, 77, 52, 23, 53, 64, 74, 33

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OFFSET

1,2

COMMENTS

Conjecture: For any positive integer n, the set {x^3 + 5*y^3 + 24*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 A273287 for a similar conjecture.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..350

EXAMPLE

a(2) = 2 since 0^3 + 5*0^3 + 24*0^3 == 2 (mod 2) with 0 + 0 + 0 = 0^3, and 0^3 + 5*0^3 + 24*1^3 == 2 (mod 2) with 0 + 0 + 1 = 1^3.

a(3) = 1 since 0^3 + 5*1^3 + 24*0^3 == 2 (mod 3) with 0 + 1 + 0 = 1^3.

a(4) = 1 since 3^3 + 5*3^3 + 24*2^3 == 2 (mod 4) with 3 + 3 + 2 = 2^3.

MATHEMATICA

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]

In[2]:= Do[r=0; Do[If[Mod[x^3+5y^3+24z^3-2, 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

Cf. A000578, A273287.