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<DIV><FONT face=Arial size=2>A better bound is</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>a(18m) <= 11 * 101^2 * 1000001^2 *
(10^18+9)^(m-1)</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>The right hand side is an 18m-digit number for 1
<= m <= 243040916832487184.</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>On the other hand, under generous assumptions about
the size of prime gaps, we have</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>a(2^m) <=
PROD nextPrime(10^(2^k)).</FONT></DIV>
<DIV><FONT face=Arial
size=2>
0 <= k < m</FONT></DIV>
<DIV><FONT face=Arial size=2></FONT> </DIV>
<DIV><FONT face=Arial size=2>where the right side has 2^m digits, which would
provide an infinitude of numbers with precisely one divisor of every possible
length.</FONT></DIV>
<BLOCKQUOTE
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<DIV style="FONT: 10pt arial">----- Original Message ----- </DIV>
<DIV
style="BACKGROUND: #e4e4e4; FONT: 10pt arial; font-color: black"><B>From:</B>
<A title=mymontain@yahoo.com href="mailto:mymontain@yahoo.com">Farideh
Firoozbakht</A> </DIV>
<DIV style="FONT: 10pt arial"><B>To:</B> <A title=njas@research.att.com
href="mailto:njas@research.att.com">N. J. A. Sloane</A> ; <A
title=joshua.zucker@gmail.com href="mailto:joshua.zucker@gmail.com">Joshua
Zucker</A> ; <A title=jvospost3@gmail.com
href="mailto:jvospost3@gmail.com">Jonathan Post</A> ; <A
title=davidwwilson@comcast.net href="mailto:davidwwilson@comcast.net">David
Wilson</A> ; <A title=franktaw@netscape.net
href="mailto:franktaw@netscape.net">franktaw@netscape.net</A> ; <A
title=arndt@jjj.de href="mailto:arndt@jjj.de">Joerg Arndt</A> ; <A
title=seqfan@ext.jussieu.fr
href="mailto:seqfan@ext.jussieu.fr">seqfan@ext.jussieu.fr</A> </DIV>
<DIV style="FONT: 10pt arial"><B>Sent:</B> Thursday, December 14, 2006 10:58
AM</DIV>
<DIV style="FONT: 10pt arial"><B>Subject:</B> Re: Re : n - digit numbers with
n factors, each with a different number of digits</DIV>
<DIV><BR></DIV>
<DIV>Dear Neil and seqfans,</DIV>
<DIV> </DIV>
<DIV>Instead " 4 " in the following statement, </DIV>
<DIV> </DIV>
<DIV>
" In fact if 4 < m <= 3000 then a(18m) exists
and </DIV>
<DIV>
a(18m) < 11^2*1009^5*(10^18+9)^(m-1)+1" </DIV>
<DIV> </DIV>
<DIV>we can write " -1 ".</DIV>
<DIV> </DIV>
<DIV>
<DIV>I would like to know, what is the largest number k such that
we can write instead " 3000 " </DIV>
<DIV>in the statement.</DIV>
<DIV> </DIV>
<DIV>Best wishes,</DIV>
<DIV>Farideh.</DIV></DIV></BLOCKQUOTE></BODY></HTML>