More about powerful numbers

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 (3)root n(2) <= u(n) <= n(2)/c(2) + 0.5n (3)root n(2) (for c = zeta(3/2)/zeta(3) and n >= 170), u(n+1) - u(n) <= n (for n >= 1316), and u(n+1) - u(n) <= 4n (for n >= 1) We also study the convergence of some number series, drawing information about, the asymptotic behavior of (u(n+1) - u(n))(n).