Counting Divisors

★ 7/10number theory Counting sequences Estimation Telescoping

Problem

For $n \in \mathbb{N}^*$ , let $f(n)$ be the number of positive divisors of $n$ ; for example $f(1) = 1$ and $f(4) = 3$ . Determine, with proof, which of the following are true:

1. $f(6) = 4$ ; 2. $f(2025) = 20$ ; 3. $\displaystyle\sum_{k=1}^{2025} \frac{1}{f(6^k)} < 1$ ; 4. with $b_n = (\sqrt{2})^n$ ,

\sum_{k=1}^{2025} \frac{(-1)^{f(k+1)}}{b_{2k-1} b_{2k}} \leqslant \frac{5\sqrt{2}}{16}.

Answer

Solution

Difficulty	7/10
Topics	number theory, Counting, sequences, Estimation, Telescoping

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