# [seqfan] Re: In the vein of 103314

Allan Wechsler acwacw at gmail.com
Wed Dec 20 20:59:07 CET 2017

```I think A(5) is at most 126 -- maybe it *is* 126.
A(6) is almost certainly 127.

If A(5) = 126, then this sequence is not in OEIS.
If A(6) = 127, then this sequence is not in OEIS no matter what A(5) is.

On Wed, Dec 20, 2017 at 2:43 PM, David Wilson <davidwwilson at comcast.net>
wrote:

> Pretty sure this is wrong.
>
> There is 1 sum of 1 1st root of 1, namely {1}.
> There are 3 sums of 2 2nd roots of 1, namely {-2, 0, 2} = {-1 + -1, -1 +
> 1, 1 + 1}
> So your sequence should definitely start (1, 3, ...)  and if I counted
> right (1, 3, 10, 25, ...)
>
> The set of sums of n nth-roots includes 0 together with nonzero sums that
> exhibit n-fold symmetry about the origin of the complex plane.
> We should therefore have
>
>         a(n) == 1 (mod n)
>
> which your sequence below does not obey.
>
>
> > -----Original Message-----
> > From: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] On Behalf Of Wouter
> > Meeussen
> > Sent: Wednesday, December 20, 2017 12:58 PM
> > To: Sequence Fanatics Discussion list
> > Subject: [seqfan] Re: In the vein of 103314
> >
> > how many different absolute values can we get from the set of n-th roots
> of
> > 1?
> > I count (for n=1 .. 12)
> >
> > Table[Tally[Sort[Flatten[Table[dist/@ListNecklaces[n, Join[1 +
> 0*Range[i],
> > 0*Range[n - i]],Dihedral], {i, 1,Floor[ n/2]}]]]]//Length,{n,12}]
> >
> > {0, 1, 1, 3, 3, 4, 8, 10, 17, 18, 62, 24}
> >
> > Wouter
> >
> >
> > -----Original Message-----
> > From: Max Alekseyev
> > Sent: Wednesday, December 20, 2017 4:42 PM
> > To: Sequence Fanatics Discussion list
> > Subject: [seqfan] Re: In the vein of 103314
> >
> > They all must be distinct as otherwise the n-th primitive root would be
> a zero
> > of a polynomial of degree n-1.
> > Hence, there are binomial(k+n-1,n-1) distinct values of k-term sums of
> nth
> > roots of 1, for any k>=1.
> >
> > Regards,
> > Max
> >
> > On Tue, Dec 19, 2017 at 4:24 PM, David Wilson <davidwwilson at comcast.net>
> > wrote:
> >
> > > How many distinct values are taken on by a sum of n nth roots of 1?
> > >
> > >
> > > --
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> > >
> >
> > --
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> >
> >
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>
>
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>
```