[seqfan] Re: Q: cyclotomic cosets A006694

[Vladimir Shevelev]

I add the following:

1) A006694(n)=A037226(n) iff 2n+1 is prime;
2) Only case, when A006694(n)<A037226(n) is n=0;
3) If {C_i}. i=1,...,A006694(n), is the set of all cyclotomic cosets of 2 mod (2n+1), then
LCM(|C_1|,...,|C_{A006694(n)}|)=A002326(n).

Regards,
Vladimir

----- Original Message -----
From: Joerg Arndt <arndt at jjj.de>
Date: Tuesday, January 18, 2011 18:17
Subject: [seqfan] Re: Q: cyclotomic cosets A006694
To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>

> Now reworded to:
> 
>   a(n) is not in general the number of orbits in 
> (Z/(2n+1)Z)* generated
>   by 2, these are counted in A037226. - M. F. Hasler, Jan 
> 18 2011.
> 
> Hope that's OK.
> 
> And yes, some sequences cold benefit from definitions...
> 
> 
> * Vladimir Shevelev <shevelev at bgu.ac.il> [Jan 18. 2011 17:10]:
> > Yes, when I for the first time have seen this sequence, I did 
> not quite understand the definition. Before (up to 2008), I did 
> not see term "coset". Besides, I thought that the definition is 
> the first statement in Comment: The number of cycles in 
> permutations constructed from siteswap juggling patterns 1, 123, 
> 12345, 1234567, etc.... 
> > I did some experiments and posed this conjecture. Later I 
> understood that  it is the definition and many timis used 
> term "coset", but forgot about this my "conjecture"-this word it 
> should be removed.
> >  
> > Regards,
> > Vladimir
> > 
> > 
> > ----- Original Message -----
> > From: Joerg Arndt <arndt at jjj.de>
> > Date: Tuesday, January 18, 2011 16:09
> > Subject: [seqfan] Re: Q: cyclotomic cosets A006694
> > To: seqfan at list.seqfan.eu
> > 
> > > * Joerg Arndt <arndt at jjj.de> [Jan 18. 2011 15:04]:
> > > > In A006694 there is the comment:
> > > > "Conjecture: a(n) is the number of orbits
> > > >   in (Z\(2n+1)Z)* generated by 2."
> > > > 
> > > > But that is the definition of the sequence, correct?
> > > > (note to myself: if so, A139035 needs editing as well)
> > > > 
> > > 
> > > M. F. Hasler took care of that, thanks!
> > > 
> > > > [...]
> > > 
> > > _______________________________________________
> > > 
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > > 
> > 
> >  Shevelev Vladimir‎
> > 
> > _______________________________________________
> > 
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> 

 Shevelev Vladimir‎