# [seqfan] A214686

israel at math.ubc.ca israel at math.ubc.ca
Fri Jul 27 04:10:03 CEST 2012

I recently submitted a new sequence, A214686. This is defined so that, for
n >= 2, a(n) is the greatest integer x such that gcd(x,n!) = 1 and x/n! < 1
- sum_{j=2}^{n-1} a(j)/j!. It is not hard to show that sum_{j=2}^infinity
a(j)/j! = 1: my proof uses the fact that for any epsilon > 0, if \$n\$ is
large enough there is a prime p with n < epsilon n!/2 < p < epsilon n!.
This came up in a discussion on MathOverflow.net
http://mathoverflow.net/questions/103129/irrationality-proof-technique-no-factorial-in-the-denominator/103136#103136
as a counterexample to the conjectured rationality of sums of convergent
series with terms of the form c(n)/n! in lowest terms.

Since gcd(a(n),n!) = 1, for each n we have either a(n) = 1 or a(n) > n, and
indeed both of these cases are well represented in the data. An interesting
pattern came to light when I plotted the terms. Most of those a(n) > n are
not too much bigger than n, maybe up to 40000 when n <= 1000. But for those
n >= 6 for which a(n-1) = 1 (of which there are 166 up to n=1000), a(n)
seems to be very close to (n-1)^2. Can anyone explain this pattern?

Robert Israel
University of British Columbia