Enumeration formulas using partitions

[Christian G. Bower]

sequence to another. It is not the algorithm used, but the mapping itself.
so it is just another expression (or algorithm) for the same transform.
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Date: Thu, 8 May 2008 00:36:36 -0700
From: "Max Alekseyev" <maxale at gmail.com>
To: koh <zbi74583 at boat.zero.ad.jp>
Subject: Re: RE : Edited A137606
Cc: seqfan at ext.jussieu.fr
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2008/4/30 koh <zbi74583 at boat.zero.ad.jp>:

>    I found some property of the sequence.
>
>    The terms are not 2 Mod 3.
>                  not 1 Mod 4.
>                  not 3 Mod 5
>                  not 4 Mod 7
>                  not 6 Mod 11
>                  not 9 Mod 17
>
>    Is it correct?
>    I don't understand well the reason.

You probably meant that there are no terms equal 2 mod 3, except the
term 2; there are no terms equal 1 modulo 4, except the term 1; there
are no terms equal 3 mod 5, except the term 3 etc.
This corrected statement is true. The reason is that the sequence
A137606 admits an alternative close-form definition:

A137606 contains 1 and numbers m>1 such that either d=m-1 and m is
even, or d=2(m-1), where d is the multiplicative order of -2 modulo
2m-1.

 From this definition it follows that for m>1 the number p=2m-1 must be
prime since there exists an element with the multiplicative order p-1
modulo p: namely, this element is 2 or -2 (depending on the above
former/latter case).

Therefore,
if  m=2 Mod 3 then 2m-1 is divisible by 3, implying that 2m-1=3 and m=2;
if m=3 Mod 5 then 2m-1 is divisible by 5, implying that 2m-1=5 and m=3;
etc. etc.

I will send an update to A137606 soon.

Regards,
Max




Dear Seqfans,  I think the idea of a wiki about the OEIS
is a terrible one.  It would inevitably produce two
versions of sequences, and it would make my job even harder
than it is now.


One argument that was mentioned was that there one
could give discussions of ambiguous terms (such as
proper divisor).  But that can be handled by entries
in the Index.  I'm adding an entry for 
divisor, proper
now