Recurring decimals

[franktaw at netscape.net]

A051628 is easy to deal with.  The average value is sum 1/2^k + sum 1/5^k - sum 1/10^k, where all sums are for k>0.  This equals 1/(2-1) + 1/(5-1) - 1/(10-1) = 41/36.  The worst case is for n = 2^k, where the value is k (i.e., log_2 n).
 
Incidently, some of the terms of A051628 were incorrect.  I have sent in the following correction and extension:
 
%I A051628
%S A051628 0,1,0,2,1,1,0,3,0,1,0,2,0,1,1,4,0,1,0,2,0,1,0,3,2,1,0,2,0,1,0,5,0,1,1,2,0,1,0,3,
0,1,0,2,1,1,0,4,0,2,0,2,0,1,1,3,0,1,0,2,0,1,0,6,1,1,0,2,0,1,0,3,0,1,2,2,0,1,0,4,
0,1,0,2,1,1,0,3,0,1,0,2,0,1,1,5,0,1,0,2,0,1,0,3,1,1,0,2,0,1,0,4,0,1,1,2,0,1,0,3
%N A051628 Number of digits in decimal expansion of 1/n before the periodic part 
begins.
%O A051628 1
%K A051628 ,base,easy,nice,nonn,
%A A051628 Frank Adams-Watters (FrankTAW at Netscape.net), May 05 2006

A051626 is harder to deal with.  The worst case occurs when n is a prime, and 10 is a primitive root of that prime; in this case a(n) = n-1.  By Artin's (second) conjecture, this happens infinitely often; indeed with density C_{Artin} = 0.3739558136....  Artin's conjecture follows from the Generalized Riemann Hypothesis, and is almost certainly true.
 
Experimentally, it appears that the average value of A051626 is proportional to n^k, were k is about 0.86.  I don't have a great deal of confidence in this, however; such expressions  involving primes often have log log n terms, which are not easily detected experimentally.
 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
 
 
-----Original Message-----
From: Rob Arthan <rda at lemma-one.com>
To: seqfan mailing list <seqfan at ext.jussieu.fr>
Sent: Fri, 5 May 2006 17:09:10 +0100
Subject: Recurring decimals


Dear All,

I am interested in the efficiency of representing rational numbers as 
recurring decimals. Two relevant sequences are A51626 and A51628. Can anyone 
give me any information about the asymptotic behaviour of sum_1^k a(n) for 
these sequences - or any other information about how good or bad the 
recurring decimal representation is in practice.

Regards,

Rob.
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