Programming exercise - Fractions madness

Consider the sequence of fractions:
$F=\{\frac{23}{95},\frac{57}{23},\frac{17}{39},\frac{130}{17},\frac{11}{14},\frac{35}{11},\frac{19}{13},\frac{1}{19},\frac{35}{2},\frac{13}{7},\frac{7}{1}\}$

We define the following algorithm (n is an integer):

func(n){
  i = 0;
  Start:
    j = 0;
    while (F[j]*n not integer) {
        j++;
    }
    i++;
    n = n*F[j]
    if (n has only factors 2 and 3){
        print(i);
    }
    goto Start;
}

The first values func(6) yield are:
- 10, 24, 46, ...

Find the 100th value

You are given the 20-th element: 185500

[My timing: < 100 ms]