Number of divisors

Builds on Divisor function Divisors of a natural number

Definition

For a positive integer $n$ , the number of divisors of $n$ is $d(n) = \sigma_0(n) = \# D(n)$ . Here $\sigma_0$ is the divisor function with $k = 0$ , and $D(n)$ is the set of divisors of $n$ .

Remarks

Other common notations are $\tau(n)$ and $\sigma_0(n)$ . If $n = p_1^{e_1} \cdots p_r^{e_r}$ with distinct primes $p_i$ , then $d(n) = (e_1 + 1) \cdots (e_r + 1)$ . For example, $12 = 2^2 \cdot 3$ , so $d(12) = 3 \cdot 2 = 6$ . The number $d(n)$ is odd if and only if $n$ is a perfect square.

Reading path

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Arithmetic function definition
Divisibility definition
Divisors of a natural number definition
Divisor function definition
Number of divisors definition

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1 page builds on this one.

Lambert series

Tag	`0BKJT6WG96`
URI	`m2://0BKJT6WG96`
Aliases	`divisor-counting-function`, `tau-function`
Kind	definition
Writing	W1 stated
MSC	11A25
Defines	Number of divisors
Uses	Divisor function, Divisors of a natural number

Number of divisors

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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