[seqfan] Re: In the vein of 103314

[israel at math.ubc.ca]

I get the following values for A(1)..A(10):
  1, 3, 10, 25, 126, 127, 1716, 2241, 18469, 15231

using the following Maple program.

nexti:= proc(i,N) local ip,j,k;
  ip:= i;
  for k from N to 1 by -1 while i[k]=N-1 do od;
  if k=0 then return NULL fi;
  ip[k]:= ip[k]+1;
  for j from k+1 to N do ip[j]:= ip[k] od;
  ip
end proc:
f:= proc(N) local S, i,P,z;
   S:= {}:
   i:= <(0$N)>:
   P:= numtheory:-cyclotomic(N,z);
   while i <> NULL do
     S:= S union {rem(add(z^i[k],k=1..N),P,z)};
     i:= nexti(i,N);
   od;
   nops(S);
end proc:
seq(f(N),N=1..10);

Cheers,
Robert



On Dec 20 2017, Allan Wechsler wrote:

>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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