In this paper we study some properties of non powerful numbers. We evaluate the $n$-th non powerful number and prove for the sequence of non powerful numbers some theorems that are related to the sequence of primes: Landau, Mandl, Scherk. Related to the conjecture of Goldbach, we prove that every positive integer $ge3$ is the sum between a prime and a non powerful number.

In this paper we make an evaluation for the number of primes between two consecutive cubes, if we assume the Riemann hypothesis.

Let $p_n$ be the $n$-th prime number and $x_n=p_{n+1}^{n+1}/p_n^n$. We show that the sequence $(x_n)_{nge 1}$ is not monotone and that the series $sumlimits_{n=1}^infty 1/x_n$ is divergent. Related series are studied as well.

For $nge 1$ let $p_n$ be the $n$-th prime number and $q_n=frac{p_{n+1}}{p_n}$ for $nge 1$. Using several results of ErdH{o}s we study the sequence $(q_n)_{nge 1}$ and we prove similar results as for the sequence $(d_n)_{nge 1}$, $d_n=p_{n+1}-p_n$. We also consider the sequence $x_n=q_n^n$ for $nge 1$ and denote by $G_n$ and $A_n$ its geometrical and arithmetical averages. We prove that $1

For every powerful number $k$, let $c_k$ be the least positive integer such that $kc_k$ is a square. We give asymptotic estimations for several series whose terms depend on the sequence $(c_k)_{kgeq 1}$, from which we mention $sumlimits_{kleq x} c_k=frac{3}{pi^2}zetaleft(frac{4}{3} ight)sqrt[3]{x^2}+O(sqrt{x}ln x)$.

For every positive integer $m$ let $b_m$ be the least positive integer such that $mb_m$ is a square. We show that $limsuplimits_{n ightarrow infty}(b_{n+1}-b_n)=+infty$, $liminflimits_{n ightarrowinfty}(b_{n+1}-b_n)=-infty$ and $sumlimits_{i=1}^nfrac{1}{b_i}=sqrt{n}frac{zeta(3/2)}{zeta(3)}+O(ln n)$.