TY - JOUR

T1 - On the square-free parts of ⌊en!⌋

AU - Luca, Florian

AU - Shparlinski, Igor E.

N1 - 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.

PY - 2007/5

Y1 - 2007/5

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=34547852283&partnerID=8YFLogxK

U2 - 10.1017/S0017089507003734

DO - 10.1017/S0017089507003734

M3 - Article

AN - SCOPUS:34547852283

SN - 0017-0895

VL - 49

SP - 391

EP - 403

JO - Glasgow Mathematical Journal

JF - Glasgow Mathematical Journal

IS - 2

ER -