Divisor Count of 720

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Example of Use of Divisor Count Function

$\map {\sigma_0} {720} = 24$

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:

$720 = 2^4 \times 3^2 \times 5$

Thus:

\(\ds \map {\sigma_0} {720}\) \(=\) \(\ds \map {\sigma_0} {2^4 \times 3^2 \times 5^1}\)
\(\ds \) \(=\) \(\ds \paren {4 + 1} \paren {2 + 1} \paren {1 + 1}\)
\(\ds \) \(=\) \(\ds 30\)


The divisors of $720$ can be enumerated as:

$1$, $2$, $3$, $4$, $5$, $6$, $8$, $9$, $10$, $12$, $15$, $16$, $18$, $20$, $24$, $30$, $36$, $40$, $45$, $48$, $60$, $72$, $80$, $90$, $120$, $144$, $180$, $240$, $360$, $720$

This sequence is A018609 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$