Divisor Count of 30

From ProofWiki
Jump to navigation Jump to search

Example of Use of Divisor Counting Function

$\map {\sigma_0} {30} = 8$

where $\sigma_0$ denotes the divisor counting function.


Proof

From Divisor Counting Function from Prime Decomposition:

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

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:

$30 = 2 \times 3 \times 5$

Thus:

\(\ds \map {\sigma_0} {30}\) \(=\) \(\ds \map {\sigma_0} {2^1 \times 3^1 \times 5^1}\)
\(\ds \) \(=\) \(\ds \paren {1 + 1} \paren {1 + 1} \paren {1 + 1}\)
\(\ds \) \(=\) \(\ds 8\)


The divisors of $30$ can be enumerated as:

$1, 2, 3, 5, 6, 10, 15, 30$

This sequence is A018255 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$