On the square-free parts of ⌊en!⌋

Florian Luca; Igor E. Shparlinski

doi:10.1017/S0017089507003734

On the square-free parts of ⌊en!⌋

Florian Luca^*, Igor E. Shparlinski

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

2 Citations (Scopus)

26 Downloads (Pure)

Abstract

In this note, we show that if we write ⌊en!⌋ = s(n)u(n) ², where s(n) is square-free then S(N)=∏_n≤NS(n) has at least C log log N distinct prime factors for some absolute constant C > 0 and sufficiently large N. A similar result is obtained for the total number of distinct primes dividing the with power-free part of s(n) as n ranges from 1 to N, where m > 3 is a positive integer. As an application of such results, we give an upper bound on the number of n < N such that ⌊en!⌋ is a square.

Original language	English
Pages (from-to)	391-403
Number of pages	13
Journal	Glasgow Mathematical Journal
Volume	49
Issue number	2
DOIs	https://doi.org/10.1017/S0017089507003734
Publication status	Published - May 2007

Bibliographical note

Copyright 2007 Cambridge University Press. Article originally published in Glasgow Mathematical Journal, Volume 49, Issue 2, pp. 391-403. The original article can be found at http://dx.doi.org/10.1017/S0017089507003734.

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abstract = "In this note, we show that if we write ⌊en!⌋ = s(n)u(n) 2, where s(n) is square-free then S(N)=∏n≤NS(n) has at least C log log N distinct prime factors for some absolute constant C > 0 and sufficiently large N. A similar result is obtained for the total number of distinct primes dividing the with power-free part of s(n) as n ranges from 1 to N, where m > 3 is a positive integer. As an application of such results, we give an upper bound on the number of n < N such that ⌊en!⌋ is a square.",

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note = "Copyright 2007 Cambridge University Press. Article originally published in Glasgow Mathematical Journal, Volume 49, Issue 2, pp. 391-403. The original article can be found at http://dx.doi.org/10.1017/S0017089507003734.",

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AU - Shparlinski, Igor E.

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On the square-free parts of ⌊en!⌋

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