Divisor Count of 6

Example of Use of Divisor Count Function

$\map {\sigma_0} 6 = 4$

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:

$6 = 2 \times 3$

Thus:

\(\ds \map {\sigma_0} 6\)

\(=\)

\(\ds \map {\sigma_0} {2^1 \times 3^1}\)

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

\(\ds 4\)

The divisors of $6$ can be enumerated as:

$1, 2, 3, 6$

$\blacksquare$

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): divisor function