[seqfan] A214686

[israel at math.ubc.ca]

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