[seqfan] Correspondent Uri Even-Chen needs help

[N. J. A. Sloane]

Dear Seq Fans,   I received the following message,
which I do not have time to answer myself. 
Could someone help him?  The questions seem just to
involve elementary number theory.
Thanks,  Neil

>From urievenchen at gmail.com Thu Jun 10 18:59:01 2010
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>Sender: urievenchen at gmail.com
>Date: Fri, 11 Jun 2010 01:58:54 +0300
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>Subject: The On-Line Encyclopedia of Integer Sequences - 2, 6, 18, 54, 162,   486 ...
>From: Uri Even-Chen <uri at speedy.net>
>To: "N. J. A. Sloane" <njas at research.att.com>,         "David W. Wilson" <davidwwilson at comcast.net>

>To N. J. A. Sloane and David W. Wilson,

>I was searching The On-Line Encyclopedia of Integer Sequences for the
>sequence of the numbers above, representing the length of the period
>of the binary representation of 1/3, 1/9, 1/27, 1/81 etc. (1/3^n).  I
>checked a few numbers and it was always 2*(3^(n-1)), but I don't have
>a proof that all the sequence is 2*(3^(n-1)).

>I am also interested in length of the period in base-b for all
>integers, especially when the period is n-1, which occurs only for
>prime numbers (but not all prime numbers) for any base b.  There are a
>few integer sequences, I would like to know if you have them:

>1. the numbers for which period is n-1 for any base b (b=2,3,4...10...)
>2. the period itself for any base b (for n=1, n=2, n=3, n=4 etc.)

>for any base there are 2 sequences above, so the number of sequences
>is infinite.  Do you have sequences which are related to some specific
>argument, in this example the base b?

>Also, I would like to know if you have sequences for pitagoras
>numbers, for example 3,4,5 ; 5,12,13 etc.  And how do you represent
>sequences of 3 numbers?

>Other sequences may apply too, for example the length of the period of
>1/7^n or 1/5^n in any base b.  For example, if using 1/7^n and base
>10, the sequence is 6, 42, 294, 2058, 14406 ... again, I don't know if
>it's always (7^n - 7^(n-1)) (or 6*(7^(n-1))).  Also the binary
>sequence of 1/7^n is 3, 21, 147, 1029, 7203, 50421 ... but I don't
>know if it's always 3*(7^(n-1)).  Also the binary sequence of 1/5^n is
>4, 20, 100, 500, 2500, 12500, 62500.... but I don't know if it's
>always 4*(5^(n-1)).

>Thanks,
>Uri Even-Chen
>Mobile Phone: +972-50-9007559
>E-mail: uri at speedy.net
>Blog: http://www.speedy.net/uri/blog/