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  Volume 5, Issue 3, Article 62
Asymptotic Formulae

    Authors: Rafael Jakimczuk,  
    Keywords: Primes, Powers of primes, Cipolla's expansion.  
    Date Received: 09/01/04  
    Date Accepted: 08/06/04  
    Subject Codes:

11N05, 11N37.

    Editors: Sever S. Dragomir,  

Let $ t_{s,n} $ be the $ n$-th positive integer number which can be written as a power $ p^t$, $ tgeq s$, of a prime $ p$ ($ sgeq 1$ is fixed). Let $ % pi _s(x)$ denote the number of prime powers $ p^t$, $ tgeq s$, not exceeding $ x$. We study the asymptotic behaviour of the sequence $ t_{s,n} $ and of the function $ pi _s(x)$. We prove that the sequence $ t_{s,n} $ has an asymptotic expansion comparable to that of $ p_n $ (the Cipolla's expansion).

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