Let $k$ be a fixed integer. We study the asymptotic formula of $R(H,r,k)$, which is the number of positive integer solutions $1\le x,y,z\le H$ such that the polynomial $x^2+y^2+z^2+k$ is $r$-free. We obtained the asymptotic formula of $R(H,r,k)$ for all $r\ge 2$. Our result is new even in the case $r=2$. We proved that $R(H,2,k)=c_k{H}^3+O\left(H^{9/4+\epsilon }\right)$, where $c_k>0$ is a constant depending on $k$. This improves upon the error term $O\left(H^{7/3+\epsilon }\right)$ obtained by G.-L. Zhou, Y. Ding (2022).

The main objective of this paper is to analyze the unimodal character of the frequency function of the largest prime factor. To do that, let P(n) stand for the largest prime factor of n. Then define f(x,p): = #{n ≤ x | P(n) = p}. If f(x,p) is considered as a function of p, for 2 ≤ p ≤ x, the primes in the interval [2,x] belong to three intervals I₁(x) = [2,v(x)], I₂(x) = ]v(x),w(x)[ and I₃(x) = [w(x),x], with v(x) < w(x), such that f(x,p) increases for p ∈ I₁(x), reaches its maximum value in...