Divisor Count of 1680

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Example of Use of Divisor Count Function

$\map {\sigma_0} {1680} = 40$

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:

$1680 = 2^4 \times 3 \times 5 \times 7$


Thus:

\(\ds \map {\sigma_0} {1680}\) \(=\) \(\ds \map {\sigma_0} {2^4 \times 3^1 \times 5^1 \times 7^1}\)
\(\ds \) \(=\) \(\ds \paren {5 + 1} \paren {1 + 1} \paren {1 + 1} \paren {1 + 1}\)
\(\ds \) \(=\) \(\ds 40\)


The divisors of $1680$ can be enumerated as:

$1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 16, 20, 21, 24, 28, 30, 35, 40, 42, 48, 56, 60,$
$70, 80, 84, 105, 112, 120, 140, 168, 210, 240, 280, 336, 420, 560, 840, 1680$

This sequence is A178878 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$