Finding a conjecture relating prime numbers to divisors
Hello I was just wondering if anyone could help me find a conjecture, theorem, hypothesis around this particular result.
If p,q are prime numbers, m,n are integers, and f(r) is the number of nontrivial divisors of r (ie. not 1 or r itself).
That if r=(p^m)*(q^n) (p≠q) that f(r)=(m+1)n+(m-1)
Thanks, I just really suck at looking for previous groundwork involving this.
Re: Finding a conjecture relating prime numbers to divisors
Since p and q are unique primes, every factor of r must be a product of some number of p's and some number of q's. A given factor of r can have anywhere between 0 p's in it and m p's in it. Likewise, it can have between 0 q's and n q's.
You can either use some combinatoric methods or write it out in general in a table and count them up. Consider to denote the number of M's and number of N's in a factor. Then one factor is (0,1), which respresents . Another is (0,2). Can you count them all up from there?