# number of divisors

The number of positive divisors of *n* is denoted by **d( n)** (or

**tau(**or better,

*n*)**τ(**. Here are the first few values of this function:

*n*)integer n |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

d(n) |
1 | 2 | 2 | 3 | 2 | 4 | 2 | 4 | 3 | 4 | 2 | 6 | 2 | 4 | 4 | 5 |

Clearly, for primes *p*, d(*p*)=2; and for prime powers, d(*p ^{n}*)=

*n*+1. For example, 3

^{4}has the five (4+1) positive divisors 1, 3, 3

^{2}, 3

^{3}, and 3

^{4}.

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) = (e_{1}+1)(e_{2}+1)(e_{3}+1) ... (e_{k}+1).

For example, 4200 is 2^{3}3^{1}5^{2}7^{1}, so it has
(3+1)(1+1)(2+1)(1+1) = 48 positive divisors.

Printed from the PrimePages <primes.utm.edu> © Chris Caldwell.