I am trying to get an intuitive meaning (a proof) or why the following is true. $$\sum_{k = 1}^n \sum_{d|k} 1 = \sum_{d = 1}^n \left[\frac{n}{d} \right]$$
I know that $f(x) = [x] = \sum_{n \leq x} 1$ but I can't see it in the case of a fraction.
Edit: can someone please tell me what I am doing wrong. I believe its something to do with summations. I know the divisor function $\tau(k) = \sum_{d|k} 1 $ so consider $$\sum_{n \leq x} \tau(n) = \sum_{n \leq x}\sum_{d|n} 1 = \sum_{n \leq x} \left[ \frac{x}{n} \right]$$ so this must mean $$\sum_{d|n} 1 = \left[ \frac{x}{n} \right]$$ which is clearly false. So what am I missing?