Divisor Count of 3657

Example of Use of Divisor Count Function

$\map {\sigma_0} {3657} = 8$

where $\sigma_0$ denotes the divisor count function.

Proof

From Divisor Count Function from Prime Decomposition:

$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$

where:

$r$ denotes the number of distinct prime factors in the prime decomposition of $n$

$k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.

We have that:

$3657 = 3 \times 23 \times 53$

Thus:

\(\ds \map {\sigma_0} {3657}\)

\(=\)

\(\ds \map {\sigma_0} {3^1 \times 23^1 \times 53^1}\)

\(\ds \)

\(=\)

\(\ds \paren {1 + 1} \paren {1 + 1} \paren {1 + 1}\)

\(\ds \)

\(=\)

\(\ds 8\)

The divisors of $3657$ can be enumerated as:

$1, 3, 23, 53, 69, 159, 1219, 3657$

$\blacksquare$