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

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.

Sequence in context: A283615 A172483 A216396 * A334955 A117848 A025242

Adjacent sequences: A273485 A273486 A273487 * A273489 A273490 A273491

Keyword

nonn

Author

Zhi-Wei Sun, Aug 28 2016

Status

approved