%I #9 Aug 29 2016 02:56:08
%S 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,
%T 21,25,30,14,29,27,23,25,34,29,42,19,42,35,57,16,69,45,41,23,45,43,43,
%U 34,60,77,52,23,53,64,74,33
%N 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).
%C 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.
%C See also A273287 for a similar conjecture.
%H Zhi-Wei Sun, <a href="/A273488/b273488.txt">Table of n, a(n) for n = 1..350</a>
%e 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.
%e a(3) = 1 since 0^3 + 5*1^3 + 24*0^3 == 2 (mod 3) with 0 + 1 + 0 = 1^3.
%e a(4) = 1 since 3^3 + 5*3^3 + 24*2^3 == 2 (mod 4) with 3 + 3 + 2 = 2^3.
%t CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]
%t 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}]
%Y Cf. A000578, A273287.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Aug 28 2016