##### Properties of non powerful numbers

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.

##### Relating the Riemann Hypothesis and the primes between two cubes

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

##### SOME PROPERTIES OF THE SEQUENCE OF PRIME NUMBERS

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.

##### ON THE RATIO OF CONSECUTIVE PRIMES

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