number of divisors
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000:
The hardware and software on this system was updated September 4th.  Please let me know of any problem you encounter. <>

The number of positive divisors of n is denoted by d(n) (or tau(n) or better, tau(n)). Here are the first few values of this function:

integer n 123456 78 91011121314 1516
d(n) 122324 24 342624 45

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

p1^e1 . p2^e2 . ... . pk^ek
then the number of divisors is
tau(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.

Chris K. Caldwell © 1999-2014 (all rights reserved)