G. Mincu and L.Panaitopol: More about powerful numbers, p.451-460

Abstract:

We prove in this paper stronger inequalities for the function $K(x)$ which measures the distribution of powerful numbers. We use them in order to study the sequence $(u_n)_n$ of powerful numbers, proving the inequalities $n^2/c^2+0.3n\sqrt[3]{n^2}\leq u_n\leq n^2/c^2+0.5n\sqrt[3]{n^2}$ (for $c=\zeta(3/2)/\zeta(3)$ and $n\geq170$), $u_{n+1}-u_n\leq n$ (for $n\geq1316$), and $u_{n+1}-u_n\leq 4n$ (for $n\geq1$). We also study the convergence of some number series, drawing information about the asymptotic behavior of $(u_{n+1}-u_n)_n$.

Key Words: powerful numbers, inequalities

2000 Mathematics Subject Classification: Primary: 11N25,
Secondary: 11A25.

Download the paper in pdf format here.