Divisor Counting Function/Examples/108

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Example of Use of $\tau$ Function

$\map \tau {108} = 12$

where $\tau$ denotes the $\tau$ Function.


Proof

From Tau Function from Prime Decomposition:

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

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:

$108 = 2^2 \times 3^3$

Thus:

\(\ds \map \tau {108}\) \(=\) \(\ds \map \tau {2^2 \times 3^3}\)
\(\ds \) \(=\) \(\ds \paren {2 + 1} \paren {3 + 1}\)
\(\ds \) \(=\) \(\ds 12\)


The divisors of $108$ can be enumerated as:

$1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108$

This sequence is A018287 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

$\blacksquare$