The prime factors of 30 are {2,3,5}
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.
Find more.
Other tasks:
Show that they cannot be semi-prime.
Show they must be square-free.
Well, let's prove that the prime factors cannot be a, a, b and c.
If 1/a + 1/a + 1/b + 1/c -1/a*a*b*c = 1
Then (2abc + a*a*c + a*a*b - 1) = a*a*b*c
the RHS is a multiple of a, but the LHS is not.
This is a contradiction, so the prime factors cannot be a, a, b and c.
Similarly, they cannot be a, a, and anything else.
So the product must be square-free.
Square-free (proof)
