[seqfan] Re: Correspondent Uri Even-Chen needs help
Alonso Del Arte
alonso.delarte at gmail.com
Tue Jun 15 00:07:32 CEST 2010
I can at least help with one his side questions:
A009004<http://www.research.att.com/~njas/sequences/A009004>contains
the smallest of each Pythagorean triple, while
A156681 <http://www.research.att.com/~njas/sequences/A156681> gives the
"middle" of each triple, and
A156682<http://www.research.att.com/~njas/sequences/A156682> gives
the largest of each triple. This treatment is roughly analogous to what we
do with sequences of fractions: the numerators in one sequence and the
denominators in another sequence.
Uri's other questions are indeed elementary but require at least a few
minutes of thought and scribbling, and I have to get back to work pretty
soon.
Al
On Mon, Jun 14, 2010 at 4:10 PM, N. J. A. Sloane <njas at research.att.com>wrote:
>
> 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
> >X-Google-Sender-Auth: zXpVFnW0abguQQqTy0v3lKtVEyQ
> >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/
>
>
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