Tau Function of 6

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Example of Use of $\tau$ Function

$\tau \left({6}\right) = 4$

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


Proof

From Tau Function from Prime Decomposition:

$\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$

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:

$6 = 2 \times 3$

Thus:

\(\displaystyle \tau \left({6}\right)\) \(=\) \(\displaystyle \tau \left({2^1 \times 3^1}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left({1 + 1}\right) \left({1 + 1}\right)\)
\(\displaystyle \) \(=\) \(\displaystyle 4\)


The divisors of $6$ can be enumerated as:

$1, 2, 3, 6$

$\blacksquare$