number of divisors
The number of positive divisors of n is denoted by d(n) (or tau(n) or better, τ(n). Here are the first few values of this function:
Clearly, for primes p, d(p)=2; and for prime powers, d(pn)=n+1. For example, 34 has the five (4+1) positive divisors 1, 3, 32, 33, and 34.
Since d(x) is a multiplicative function, this is enough to know d(n) for all integers n--if the canonical factorization of n is
then the number of divisors is
τ(n) = (e1+1)(e2+1)(e3+1) ... (ek+1).
For example, 4200 is 23315271, so it has (3+1)(1+1)(2+1)(1+1) = 48 positive divisors.