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Properties of non powerful numbers. (English) Zbl 1170.11034

A positive integer is called non powerful, if it is divided by a prime \(p\), but not by \(p^2\). Let \(v_1<v_2<v_3\dots\) the sequence of all non powerful numbers. The authors prove – based on a former paper – the inequalities \[ v_n>n+c\sqrt n-1,8{\root 3\of n},\;n\geq 88, \]
\[ v_n<n+c\sqrt n-{\root 3\of n},\;n\geq 1, \] an asymptotic formula \[ v_n=n+\frac{\zeta(\tfrac 32)}{\zeta(3)}\sqrt n+\frac{\zeta (\tfrac 23)}{\zeta(2)}{\root3\of n}+R(n) \] and properties related to the sequence of primes (Landau, Mandl, Scherk, Chen).

MSC:

11N64 Other results on the distribution of values or the characterization of arithmetic functions
11P32 Goldbach-type theorems; other additive questions involving primes
11N37 Asymptotic results on arithmetic functions
11A41 Primes
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