OFFSET
0,2
COMMENTS
In their abstract, Luca and Shparlinski write: "In this note, we show that if we write floor(e*n!) = s(n)*u(n)^2, where s(n) is square-free then S(N) = prod(n<=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 m-th 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 floor(e*n!) is a square."
LINKS
F. Luca and I. E. Shparlinski, On the squarefree parts of floor(e*n!), Glasgow Math. J., 49 (2007), 391-403.
PROG
(PARI) a(n) = prod(i=1, n, core(i! * polcoef(exp(x + x*O(x^i)) / (1 - x), i)))
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Marcus, Feb 27 2013
STATUS
approved