How to get smooth numbers efficiently by using dependent iterators in Table

Question

I have the following code for getting smooth numbers up to Prime[4] :
g = 100;

t1 = Table[{2^x*3^y*5^z*7^u}, {u, 0, Log[7, g]}, {z, 0, Log[5, g/(7^u)]}, {y, 0, Log[3,g/(7^u*5^z)]}, {x, 0, Log[2, g/(7^u*5^z*3^y)]}]
t2 = Sort[Flatten[t1]]

I am looking for a general solution for k Primenumbers (in ascending order) using Table which refers to iterators appearing in the loop before. I know I can use FactorInteger and Select or fast code from (Creating a Table with varying depth and interdependent limits)  to solve this problem, but how to tackle it using Table? Is there a way to do this? I think the answer lies somewhere in Table with dependent iterator  but here are too many tiwsts for my level of understanding.

Bob Hanlon · Accepted Answer

Clear["Global`*"]

g = 100;

t1 = Table[{2^x*3^y*5^z*7^u}, {u, 0, Log[7, g]}, {z, 0, Log[5, g/(7^u)]}, {y, 
    0, Log[3, g/(7^u*5^z)]}, {x, 0, Log[2, g/(7^u*5^z*3^y)]}];

t2 = Sort[Flatten[t1]]

(* {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 
30, 32, 35, 36, 40, 42, 45, 48, 49, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 
81, 84, 90, 96, 98, 100} *)

Generalizing to n primes
smooth[n_Integer?Positive, g_Integer?Positive] :=
 Module[{a, func, factors, iterators,
   primes = Prime /@ Range[n],
   vars = Array[a, n]},
  func = Times @@ (primes^vars);
  factors = #[[1]]^#[[2]] & /@ Transpose[{primes, vars}];
  iterators = {#[[1]], 0, Log[#[[2]], #[[3]]]} & /@
    Transpose[{
      Reverse@vars,
      Reverse@primes,
      g/FoldList[Times, 1, Most@Reverse@factors]}];
  Table[func, Evaluate[Sequence @@ iterators]] // Flatten // Sort // Quiet]

Checking against original calculations
smooth[4, g] == t2

(* True *)

How to get smooth numbers efficiently by using dependent iterators in Table

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