Sum of all elements of a Set

Question

Let's say I want to determine the number of natural numbers for an $x \in N$ this particular way: $$f(x) = \sum_{i=1}^x a\in\lbrace 1 : x\space\mathbf {mod}\space i=0\rbrace$$Is this the correct way to express this? Thanks in advance.

Answer 1

If you want $f(x)$ to be the result of adding $1$ for each $1\leq i\leq x$ such that $x\equiv 0\bmod i$, then you are counting the number of divisors of $x$, and it is usually denoted $d(x)$ or $\sigma_0(x)$.

One is allowed to be much more expressive in the range of a sum than just the numbers from a start to a finish value. It's not uncommon to see expressions like $$\sum_{\large \substack{(x,y)\in A\cap B\\ x<5\\ \sin(y)<0}}\cdots$$ Therefore a much easier way of writing what you want is $$\sum_{d\mid n}$$ where the notation $d\mid n$ means "$d$ divides $n$".

See this Wikipedia article for more.