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 -