n,x   - output
2,1   - 0   (since CD(2,1)={1,3}, distinct prime divisors={}, ravenity=0)
5,1   - 2
100,4 - 5
720,6 - 11

ŒRḟ0*3+µÆfFœ-µQL

ŒRḟ0*3+µÆfFœ-µQL  Main link. Arguments, x, n

ŒR                Range; yield [-x, ..., x].
  ḟ0              Filter out 0.
    *3            Cube each remaining integer.
      +           Add n to all cubes.
       µ          Begin a new, monadic link. Argument: A (list of sums)
        Æf        Factorize each k in A.
          F       Flatten the resulting, nested list.
           œ-     Perform multiset difference with A.
                  If k in A is prime, Æf returns [k], adding on k too many to the
                  flat list. Multiset difference with A removes exactly one k from
                  the results, thus getting rid of primes.
                  If k is composite (or 1), it cannot appear in the primes in the
                  flat list, so subtracting it does nothing.
             µ    Begin a new, monadic link. Argument: D (list of prime divisors)
              Q   Unique; deduplicate D.
               L  Compute the length of the result.

l{st#mP+Q^d3s_BMSE

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

L3mD(«-Dp_Ïvyf`})Úg

:3^t_h+tZp~)"@Yf!]vun

:       % take n implicitly. Generate [1,2,...,n]
3^      % raise to 3, element-wise
t_h     % duplicate, negate, concatenate horizontally: [1,2,...,n,-1,2,...-n]
+       % take x implicitly. Add to that array
t       % duplicate
Zp      % array that contains true for primes
~       % logical negate
)       % apply index to keep only non-primes
"       % for each number in that array
  @     %   push that number
  Yf!   %   prime factors, as a column array
]       % end for each
v       % concatenate vertically all factors
u       % remove repeated factors
n       % number of elements of that array. Implicitly display

#@~.@(,@:q:-.0&,)@:+(|#^&3)@i:

f =: #@~.@(,@:q:-.0&,)@:+(|#^&3)@i:
   100 f 4
5

#@~.@(,@:q:-.0&,)@:+(|#^&3)@i:
                            i:  Range from -x to x
                    (     )@    Apply this verb to the range:
                       ^&3        a) every item cubed
                     |            b) absolute value of every item
                      #           c) every item in a) repeated b) times; this removes 0
                                     and produces some harmless duplication
                   +            Add n to every element of the resulting list
     (          )@:             Apply this verb to the resulting vector:
             0&,                  a) the vector with 0 appended
      ,@:q:                       b) flat list of prime divisors in the vector
                                     (and some extra 0s since we flatten an un-even matrix)
           -.                     c) list b) with elements of a) removed; this gets rid of
                                     the extra 0s and all primes that were in the list
#@~.@                           Remove duplicates and take length

lambda r,n:len({z for z in{v for f in{t for u in[[r-q**3,r+q**3]for q in range(1,n+1)]for t in u if any(t%g<1 for g in range(2,t))}for v in range(2,f)if f%v<1}if all(z%g>0 for g in range(2,z))})

(n,x)->omega(factorback(select(k->!isprime(k),vector(2*x,i,n+(i-(i<=x)-x)^3))))

(n,x)->omega(factorback(select(k->!isprime(k),concat(vector(x,i,n-i^3),vector(x,i,n+i^3)))))

->(n,x){require'prime'
v=((-x..x).to_a-[0]).map{|i|n+i**3}.reject{|e|Prime.prime?(e)}
Prime.each(v[-1]).select{|i|v.any?{|e|e%i==0}}.size}

a=range
p=lambda n:any(n%x<1for x in a(2,n))
r=lambda n,x:len(set(sum([[x for x in a(2,z+1)if z%x<1&1>p(x)]for z in filter(p,[n+z**3for z in a(-x,x+1)])],[])))

def is_composite(n):
    return any(n % x == 0 for x in range(2, n))

def prime_factors(n):
    return {x for x in range(2, n+1) if n % x == 0 and not is_composite(x)}

def ravenity(n, x):
    nums = [n + z**3 for z in range(-x, x+1)]
    nums = filter(is_composite, nums)
    factors = map(prime_factors, nums)
    factors = sum(factors, [])
    #remove duplicates
    factors = set(factors)
    return len(factors)

Tr[1^Union[First/@Join@@FactorInteger/@Select[z=Range@#2^3;Join@@{#-z,#+z},Not@*PrimeQ]]]&

F[n_, x_] :=
  Length[Union[                                        (* number of unique elements   *)
    First /@                                           (* drop multiplicities         *)
      Join @@                                          (* join all prime factor lists *)
        FactorInteger /@                               (* compute prime factors       *)
          Select[                                      (* select those...             *)
            Join @@ {n - Range[x]^3, n + Range[x]^3},  (* ...candidates...            *)
            Not@*PrimeQ]]]                             (* ...that are not prime       *)