Ravenity of Cube Distance Numbers
Pyth - 21 19 18 bytes
I wonder if there's a trick.
l{st#mP+Q^d3s_BMSE

Test Suite.
l                   Length
 {                  Uniquify
  s                 Combine divisor lists
   t#               Filter by if more than one element
     PM             Take prime factorization of each number
       +RQ          Add each num in list to input
          s_BM      Each num in list and its negative (with bifurcate)
              ^R3   Cube each num in list
                 SE Inclusive unary range - [1, 2, 3,... n] to input