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strage sum of number of divisors

Source: Romanian National Olympiad 2014, Grade X, Problem 1

March 2, 2019
number theory

Problem Statement

Let be a natural number n. n. Calculate ∑k=1n2#{d∈N∣1≤d≤k≤d2≤n2∧k≡0(modd)}. \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.
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