MathDB

4

Part of 2024 VJIMC

Problems(2)

Upper bound on integers satisfying n|3^n-1

Source: VJIMC 2024, Category I, Problem 4

4/14/2024
Let p>2p>2 be a prime and let A={n∈N:2p∣n and p2∤n and n∣3n−1}.\mathcal{A}=\{n \in \mathbb{N}: 2p \mid n \text{ and } p^2\nmid n \text{ and } n \mid 3^n-1\}. Prove that lim sup⁡k→∞∣A∩[1,k]∣k≤2log⁡3plog⁡p.\limsup_{k \to \infty} \frac{\vert \mathcal{A} \cap [1,k]\vert}{k} \le \frac{2\log 3}{p\log p}.
number theoryAnalytic Number TheoryDivisibility
Recurrent sequence with divisor function is unbounded, but not monotone

Source: VJIMC 2024, Category II, Problem 4

4/14/2024
Let (bn)n≥0(b_n)_{n \ge 0} be a sequence of positive integers satisfying bn=d(∑i=0n−1bk)b_n=d\left(\sum_{i=0}^{n-1} b_k\right) for all n≥1n \ge 1. (By d(m)d(m) we denote the number of positive divisors of mm.) a) Prove that (bn)n≥0(b_n)_{n \ge 0} is unbounded. b) Prove that there are infinitely many nn such that bn>bn+1b_n>b_{n+1}.
functionDivisorsSequencesIntegersnumber theory