Divisor Count of 90

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Example of Use of Divisor Count Function

$\map {\sigma_0} {90} = 12$

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:

$90 = 2 \times 3^2 \times 5$


Thus:

\(\ds \map {\sigma_0} {90}\) \(=\) \(\ds \map {\sigma_0} {2^1 \times 3^2 \times 5^1}\)
\(\ds \) \(=\) \(\ds \paren {1 + 1} \paren {2 + 1} \paren {1 + 1}\)
\(\ds \) \(=\) \(\ds 12\)


The divisors of $90$ can be enumerated as:

$1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90$

This sequence is A018278 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$