On some properties of the function of the number of relatively prime subsets of $\{1, 2, ..., n\}$

TL;DR

This paper investigates properties of the function counting relatively prime subsets of the first n natural numbers, providing inequalities and relations that advance understanding of their combinatorial and number-theoretic structure.

Contribution

It solves specific open problems related to the properties of functions counting coprime subsets and establishes new inequalities involving these functions.

Findings

Proves that f(n)^2 > f(n-k)f(n+k) for n ≥ k+1, k ≥ 2.

Establishes inequalities among ratios of g(n) for large n, revealing structural patterns.

Provides solutions to problems proposed by Prapanpong Pongsriiam.

Abstract

In the paper we solve few problems proposed by Prapanpong Pongsriiam. Let $f(n)$ denote the number of relatively prime subsets of $\{1, 2, 3, \dots, n\}$ and $g(n)$ denote the number of subsets $A$ of $\{1, 2, 3, \dots, n\}$ such that gcd$(A)>1$ and gcd$(A, n+1)=1$ . We show that $f_n^2-f_{n-k}f_{n+k}>0$ for $n\geq k+1\quad (k\geq2)$. We also show $\frac{g(6n-2)}{g(6n-4)}>\frac{g(6n)}{g(6n-2)}>\frac{g(6n+2)}{g(6n)}<\frac{g(6n+4)}{g(6n+2)}$ for large $n$.