Let $\left[x\right]$ be an integer part of $x$ and $d\left(n\right)$ be the number of positive divisor of $n$. Inspired by some results of M. Jutila (1987), we prove that for $1<c<\frac{6}{5}$, $$\sum _{n\le x}d\left(\left[n^c\right]\right)=cxlogx+(2\gamma -c)x+O\left(\frac{x}{logx}\right),$$ where $\gamma$ is the Euler constant and $\left[n^c\right]$ is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.

For a prime p > 2, an integer a with gcd(a,p) = 1 and real 1 ≤ X,Y < p, we consider the set of points on the modular hyperbola ${}_{a,p}(X,Y)=(x,y):xy\equiv a\left(modp\right),1\le x\le X,1\le y\le Y$. We give asymptotic formulas for the average values ${\sum }_{\begin{array}{c}(\end{array}x,y){\in }_{a,p}(X,Y)x\ne y*}\phi \left(\right|x-y\left|\right)/|x-y|$ and ${\sum }_{\begin{array}{c}(\end{array}x,y){\in }_{a,p}(X,X)x\ne y*}\phi \left(\right|x-y\left|\right)$ with the Euler function φ(k) on the differences between the components of points of ${}_{a,p}(X,Y)$.

Improving on some results of J.-L. Nicolas [15], the elements of the set $𝒜=𝒜(1+z+z^3+z^4+z^5)$, for which the partition function $p(𝒜,n)$ (i.e. the number of partitions of $n$ with parts in $𝒜$) is even for all $n\ge 6$ are determined. An asymptotic estimate to the counting function of this set is also given.

We show that $n^2+1$ is powerfull for $O\left(x^{2/5+ϵ}\right)$ integers $n\le x$ at most, thus answering a question of P. Ribenboim.