MathDB

FE involving number-theoretic functions under composition with f

Source: 2023 Thailand Online MO P5

February 12, 2023

number theoryNumber theoretic functionsArithmetic Functionsfunctional equation

Problem Statement

For each positive integer

k

, let

d(k)

be the number of positive divisors of

k

and

\sigma(k)

be the sum of positive divisors of

k

. Let

\mathbb N

be the set of all positive integers. Find all functions

f: \mathbb{N} \to \mathbb N

such that \begin{align*} f(d(n+1)) &= d(f(n)+1) \text{and} \\ f(\sigma(n+1)) &= \sigma(f(n)+1) \end{align*} for all positive integers

n