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<DIV>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).</DIV>
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<DIV>Incidently, some of the terms of A051628 were incorrect. I have sent in the following correction and extension:</DIV>
<DIV> </DIV>
<DIV>%I A051628<BR>%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,<BR>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,<BR>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<BR>%N A051628 Number of digits in decimal expansion of 1/n before the periodic part <BR>begins.<BR>%O A051628 1<BR>%K A051628 ,base,easy,nice,nonn,<BR>%A A051628 Frank Adams-Watters (<A href="mailto:FrankTAW%40Netscape.net">FrankTAW@Netscape.net</A>), May 05 2006<BR></DIV>
<DIV>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.</DIV>
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<DIV>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.</DIV>
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<DIV>Franklin T. Adams-Watters<BR>16 W. Michigan Ave.<BR>Palatine, IL 60067<BR>847-776-7645</DIV>
<DIV> </DIV> <BR>-----Original Message-----<BR>From: Rob Arthan <rda@lemma-one.com><BR>To: seqfan mailing list <seqfan@ext.jussieu.fr><BR>Sent: Fri, 5 May 2006 17:09:10 +0100<BR>Subject: Recurring decimals<BR><BR>
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<DIV class=AOLPlainTextBody id=AOLMsgPart_0_702fe9d3-9c75-4458-902b-4ee1eeebe82c><PRE><TT>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.
</TT></PRE></DIV><!-- end of AOLMsgPart_0_702fe9d3-9c75-4458-902b-4ee1eeebe82c --></DIV></DIV>
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