[seqfan] Re: The discriminator of a sequence
Olivier Gerard
olivier.gerard at gmail.com
Wed May 4 08:45:07 CEST 2016
Dear Neil,
The OEIS has the original sequence and a ref to the original 1985 article
for the discriminator
A016726 Smallest k such that 1, 4, 9, ..., n^2 are distinct mod k.
as well as two immediate similar sequences
A192419 <http://oeis.org/A192419> for cubes
A192420 <http://oeis.org/A192420> for fourth powers
described in a further article by other authors.
The search engine makes it difficult to find references for the
discriminator as it is
converted into the keyword "discriminant" which is quite frequent.
Here is some code in "Wolfram Language"
Clear[Discriminator];
Discriminator[kl_: {__Integer}] :=
Module[{dn, sl, mn}, dn = 1;
Table[ sl = Abs[Take[kl, i]]; mn = Max[sl];
While[dn <= mn + 1 && Length[Union[Mod[sl, dn]]] < i, dn++];
dn, {i, 1, Length[kl]}]] /; (Sort[Abs[kl]] == Union[Abs[kl]])
it computes the discriminator of a sequence of integers
(but it cannot contain any duplicate).
The Discriminator of odious numbers is
A062383 <http://oeis.org/A062383>
1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32,...
but we need to make a comment about it.
The Discriminator of evil numbers starts
1, 2, 4, 4, 7, 8, 8, 8, 13, 16, 16, 16, 16, 16, 16, 16, 31, 32,
but is not in the OEIS.
Another related transform is the compressed or reduced version of the
Discriminator
and its change index for sequences who stay with the same modulo
for quite some time.
The compressed Discriminator of odious numbers as well as (n choose 2)
is just the powers of 2 and the one for factorial is
1, 2, 3, 7, 10, 13, 31, 37, 61, 83, 127, 179, 193, 277, 383, 479, 541, 641,
877, ...
(not in the OEIS)
and for 2^n -1 or 2^n+1 is
1, 3, 5, 9, 11, 13, 19, 25, 29, 37, 53, 59, 61, 67, 83, 101, ...
and seems to be A139099 <http://oeis.org/A139099>
for 3^n-1 or 3^n+1
1, 4, 5, 7, 17, 19, 25, 29, 31, 43, 53, 79, 89, 101, ...
(not in the OEIS)
The Discriminator for integer partitions with distinct parts is
1, 3, 4, 4, 5, 6, 7, 8, 10, 12, 14, 17, 20, 24, 29, 34, 40, 48, 56,
66, 78, 91, 106, 124, 144, 167, 194, 224, 258, 298, ...
(not in the OEIS)
and the compressed one for integer partitions is
1, 2, 3, 5, 7, 11, 16, 42, 52, 68, 80, 83, 84, 101, 116
(not in the OEIS)
etc.
Olivier
On Tue, May 3, 2016 at 6:59 PM, Neil Sloane <njasloane at gmail.com> wrote:
>
> Dear Seqfans,
>
> Michel Marcus found a very nice paper on the arXiv that gives a new
> transformation of sequences: you take a sequence a(n) and compute its
> discriminator and you get a new sequence!
>
> Sajed Haque, Jeffrey Shallit, Discriminators and k-Regular Sequences,
> arXiv:1605.00092, 2016
>
> It would be interesting to see what happens when this is applied to all
our
> favorite sequences.
>
> They give a couple of examples (evil, odious numbers, etc.), but
> I don't even know if their discriminators are in the OEIS (I didn't check)
>
> They mention the OEIS, so I added it to the web page "Works Citing the
> OEIS" on our wiki.
>
> Best regards
> Neil
>
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com
> Email: njasloane at gmail.com
>
> --
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