Let be a natural number $ n. $ Calculate $$ \sum_{k=1}^{n^2}\#\left\{ d\in\mathbb{N}| 1\le d\le k\le d^2\le n^2\wedge k\equiv 0\pmod d \right\} . $$ Here, $ \# $ means cardinal.
Source: Romanian National Olympiad 2014, Grade X, Problem 1
Tags: number theory
Let be a natural number $ n. $ Calculate $$ \sum_{k=1}^{n^2}\#\left\{ d\in\mathbb{N}| 1\le d\le k\le d^2\le n^2\wedge k\equiv 0\pmod d \right\} . $$ Here, $ \# $ means cardinal.