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 $n$.
Categories: math.NT math.CO
