The Prime Glossary: number of divisors

number of divisors
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The number of positive divisors of n is denoted by d(n) (or tau(n) or better,

). Here are the first few values of this function:

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(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

= (e₁+1)(e₂+1)(e₃+1) ... (e_k+1).

For example, 4200 is 2³3¹5²7¹, so it has (3+1)(1+1)(2+1)(1+1) = 48 positive divisors.