[seqfan] Re: A214686

[Vladimir Shevelev]

Consider the sequence: b(n) is the greatest integer x<=(n-1)^2 such that gcd(x,n!)=1. It is clear that b(n) is the maximal prime <=(n-1)^2-2. If  n!(1-sum{j=2,...,n-1}a(j)/j!)<=(n-1)^2, then, evidently, 
a(n)<=b(n)<=(n-1)^2-2. If there exists an exceptional set E={n: n!(1-sum{j=2,...,n-1}a(j)/j!)>=
(n-1)^2+1},  then we always have a(n)<=(n-1)^2-2 iff in every interval [(n-1)^2+1, n!(1-sum{j=2,...,n-1}a(j)/j!)] all numbers from E are not respectively prime to n! I think that the natural conjecture is that E is empty for n>=3.
 
Regards,
Vladimir

----- Original Message -----
From: israel at math.ubc.ca
Date: Friday, July 27, 2012 4:10
Subject: [seqfan] A214686
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> 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  
> 
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> 

 Shevelev Vladimir‎