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I have a question involving a function:

$\tau(n) := $#{$d\in\Bbb{Z}>0|$d divides n}

The hash symbol before the first bracket is confusing me, I don't know what this means. And could you please give an example for if I was to substitute in a value for n. Thanks

3 Answers 3

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The hash symbol denotes the cardinality, e.g. $\#G$ denotes the cardinality of a group $G$, for example $\#(\mathbb{Z}/n\mathbb{Z})^*=\phi(n)$. Another convention for $\tau(n)$ is to write $$ \tau(n)=\sum_{d\mid n}1. $$ This is the more typical way in analytic number theory; another example is $$ \pi(x)=\sum_{p\le x} 1, $$ for the prime counting function.

Concerning your question how to compute $\tau(n)$, see this MSE-question.

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It could be the cardinality (number of elements) of the set behind the hash mark. The hash symbol is also used as abbreviation for number: link.

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    So if n was say 10, $\tau(n)$ would be equal to 4? being divisible by 1, 2, 5, and 10 – 2017-01-18
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    Yes, the number of positive divisors. – 2017-01-18
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    See this article: https://en.wikipedia.org/wiki/Arithmetic_function#.CF.83k.28n.29.2C_.CF.84.28n.29.2C_d.28n.29_.E2.80.93_divisor_sums – 2017-01-18
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It's "the number of". This if $n = 6$, the possibilities for $d$ are $1, 2, 3, 6$, so $\tau(6) = 4$.

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    Thank you that makes sense :) – 2017-01-18
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    How would I go about computing $\tau(n)$ if n was a very large number (in the tens of thousands) without manually trying every number? – 2017-01-18
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    Use the prime factorization of $n$. If $n = p_1^{e_1} \ldots p_k^{e_k}$ where $p_i$ are distinct primes, $\tau(n) = (e_1 + 1) \ldots (e_k + 1)$. – 2017-01-18
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    Or you could ask [Wolfram Alpha](http://www.wolframalpha.com/input/?i=tau(23456)). – 2017-01-18