The sum of the reciprocals is 1/2+1/3+1/5=31/30
The product of the reciprocals is 1/2*1/3*1/5=1/30
The difference of the two is 31/30-1/30=1
There aren't many numbers where the final value of this procedure is a whole number.
Show that they cannot be semi-prime.
Show they must be square-free.