Hamming numbers are numbers of the form  

    H = 2i × 3j × 5k
           where 
     i,  j,  k  ≥  0 

Hamming numbers   are also known as   ugly numbers   and also   5-smooth numbers   (numbers whose prime divisors are less or equal to 5).



Task

Generate the sequence of Hamming numbers, in increasing order.   In particular:

 Show the   first twenty   Hamming numbers.
 Show the   1691st   Hamming number (the last one below   231).
 Show the   one millionth   Hamming number (if the language – or a convenient library – supports arbitrary-precision integers).


Related tasks

 Humble numbers
 N-smooth numbers


References

 Wikipedia entry:   Hamming numbers     (this link is re-directed to   Regular number).
 Wikipedia entry:   Smooth number
 OEIS entry:   A051037   5-smooth   or   Hamming numbers
 Hamming problem from Dr. Dobb's CodeTalk (dead link as of Sep 2011; parts of the thread here and here).


package main
 
import (
    "fmt"
    "math/big"
)
 
func min(a, b *big.Int) *big.Int {
    if a.Cmp(b) < 0 {
        return a
    }
    return b
}
 
func hamming(n int) []*big.Int {
    h := make([]*big.Int, n)
    h[0] = big.NewInt(1)
    two, three, five    := big.NewInt(2), big.NewInt(3), big.NewInt(5)
    next2, next3, next5 := big.NewInt(2), big.NewInt(3), big.NewInt(5)
    i, j, k := 0, 0, 0
    for m := 1; m < len(h); m++ {
        h[m] = new(big.Int).Set(min(next2, min(next3, next5)))
        if h[m].Cmp(next2) == 0 { i++; next2.Mul(  two, h[i]) } 
        if h[m].Cmp(next3) == 0 { j++; next3.Mul(three, h[j]) } 
        if h[m].Cmp(next5) == 0 { k++; next5.Mul( five, h[k]) } 
    }
    return h
}
 
func main() {
    h := hamming(1e6)
    fmt.Println(h[:20])
    fmt.Println(h[1691-1])
    fmt.Println(h[len(h)-1])
}