The Prime Divisor Functions

Abstract

Here we consider only prime divisors of n and ask, for given order of magnitude of n, “how many prime divisors are there typically?” and “how many different ones are there?” Some of the answers will be rather counterintuitive. Thus, a 50-digit number (1021 times the age of our universe measured in picoseconds) has only about 5 different prime factors on average and — even more surprisingly — 50-digit numbers have typically fewer than 6 prime factors in all, even counting repeated occurrences of the same prime factor as separate factors.