n-smooth   numbers are positive integers which have no prime factors  >  n.

The   n   (when using it in the expression)   n-smooth   is always prime,

there are   no   9-smooth numbers.

1   (unity)   is always included in n-smooth numbers.




2-smooth   numbers are non-negative powers of two.

5-smooth   numbers are also called   Hamming numbers.

7-smooth   numbers are also called    humble   numbers.



A way to express   11-smooth   numbers is:

  11-smooth  =  2i × 3j × 5k × 7m × 11p

           where     i, j, k, m, p ≥ 0  



Task

   calculate and show the first   25   n-smooth numbers   for   n=2   ───►   n=29
   calculate and show   three numbers starting with   3,000   n-smooth numbers   for   n=3   ───►   n=29
   calculate and show twenty numbers starting with  30,000   n-smooth numbers   for   n=503   ───►   n=521   (optional)


All ranges   (for   n)   are to be inclusive, and only prime numbers are to be used.

The (optional) n-smooth numbers for the third range are:   503,   509,   and   521.

Show all n-smooth numbers for any particular   n   in a horizontal list.

Show all output here on this page.



Related tasks

   Hamming numbers
   humble numbers


References

   Wikipedia entry:   Hamming numbers     (this link is re-directed to   Regular number).
   Wikipedia entry:   Smooth number
   OEIS entry:   A000079    2-smooth numbers or non-negative powers of two
   OEIS entry:   A003586    3-smooth numbers
   OEIS entry:   A051037    5-smooth numbers or Hamming numbers
   OEIS entry:   A002473    7-smooth numbers or humble numbers
   OEIS entry:   A051038   11-smooth numbers
   OEIS entry:   A080197   13-smooth numbers
   OEIS entry:   A080681   17-smooth numbers
   OEIS entry:   A080682   19-smooth numbers
   OEIS entry:   A080683   23-smooth numbers


using Primes
 
function nsmooth(N, needed)
    nexts, smooths = [BigInt(i) for i in 2:N if isprime(i)], [BigInt(1)]
    prim, count = deepcopy(nexts), 1
    indices = ones(Int, length(nexts))
    while count < needed
        x = minimum(nexts)
        push!(smooths, x)
        count += 1
        for j in 1:length(nexts)
            (nexts[j] <= x) && (nexts[j] = prim[j] * smooths[(indices[j] += 1)])
        end
    end
    return (smooths[end] > typemax(Int)) ? smooths : Int.(smooths)
end
 
function testnsmoothfilters()
    for i in filter(isprime, 1:29)
        println("The first 25 n-smooth numbers for n = $i are: ", nsmooth(i, 25))
    end
    for i in filter(isprime, 3:29)
        println("The 3000th through 3002nd ($i)-smooth numbers are: ", nsmooth(i, 3002)[3000:3002])
    end
    for i in filter(isprime, 503:521)
        println("The 30000th through 30019th ($i)-smooth numbers >= 30000 are: ", nsmooth(i, 30019)[30000:30019])
    end
end
 
testnsmoothfilters()
 