[seqfan] Re: Name for a type of set needed

Mon Mar 27 11:33:45 CEST 2017

On Mon, Mar 27, 2017 at 12:47 AM, <seqfan-request at list.seqfan.eu> wrote:

>
>
> Message: 13
> Date: Fri, 24 Mar 2017 20:24:24 -0500
> From: "Bob Selcoe" <rselcoe at entouchonline.net>
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Subject: [seqfan] Name for a type of set needed
> Message-ID: <118E865929104ABFAE5547A3B6966BDC at OwnerPC>
> Content-Type: text/plain; format=flowed; charset="utf-8";
>         reply-type=original
>
> Hi Seqfans,
>
> I just proposed sequence A284311, titled: "Array T(n,k) read by
> antidiagonals: T(1,k) = squarefree numbers > 1 in ascending order
> (A005117(k)); columns are non-squarefree numbers (in ascending order) with
> the exact same prime factors as T(1,k)".
>
> So for example, Column 6 is the set whose members have prime factors {2,5}:
> {10, 20, 40, 50, 80, 100, 160, 200...}.
>
> Is there a standard name for such sets?
>

Hah, this is the square array I once thought about submitting! That is, the
initial column (or row, if transposed)
consists of the squarefree numbers > 1, and then what follows in the same
column (or row), are all the numbers that have
that initial number as their squarefree part.

The construction reminds me also of Clark Kimberling's dispersions, but for
that we would need to find (or construct!)
a sequence a(n) that gives consistently the next term down the line. Well,
actually, it should be easy:
a(n) = the smallest number k > n which has same squarefree part,
A007913(n), thus it should be a permutation of A013929.

I have used similar "backformation-style-dispersions" before, e.g. for
constructing A083221 "Sieve of Eratosthenes arranged as an array and read
by antidiagonals" as a "dispersion" of A250469.
But I think that in Kimberling's original terminology only dispersions of
monotonic functions were considered.

Moreover, in my opinion, for all important arrays like these, both
transpositions should be submitted in, because sooner or later (say, after
ten years, which is short time in OEIS) somebody will search for the other
orientation, and when not found, will submit their own version. It's much
better that both are in the database from the start, with other just
pointing to the other, saying "Transposition of array Axxxxxx, see comments
there."

Best regards,

Antti

>
> Cheers,
> Bob Selcoe
>
>
>
>
> ------------------------------
>
> Message: 15
> Date: Sat, 25 Mar 2017 18:09:51 -0400
> From: "M. F. Hasler" <oeis at hasler.fr>
> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Name for a type of set needed
> Message-ID:
>         <CABxCbJ0Qtv1nbqfCz6CqBhtaCnke18KJfTQDdJ+ic6xG4Nx2Jw at mail.
> gmail.com>
> Content-Type: text/plain; charset=UTF-8
>
> If you look at https://oeis.org/A284311/table you notice that there's an
> error
> in the data, leading to a value of 10 instead of 64.
>
> IMHO it would be more natural to list the numbers with given prime factors
> as rows, i.e., the transposed table, s.t. the first *column* would be
> A005117 <https://oeis.org/A005117>,
> in analogy with the similar oeis.org/A182944 <https://oeis.org/A182944> .
>
> I don't know a name for the set of numbers which has a given squarefree
> kernel (A007947), but one could simply say it's the inverse image
> A007947^(-1) of that squarefree number.
>
> Maximilian
>
> On Fri, Mar 24, 2017 at 9:24 PM, Bob Selcoe <rselcoe at entouchonline.net>
> wrote:
>
> > Hi Seqfans,
> >
> > I just proposed sequence A284311, titled: "Array T(n,k) read by
> > antidiagonals: T(1,k) = squarefree numbers > 1 in ascending order
> > (A005117(k)); columns are non-squarefree numbers (in ascending order)
> with
> > the exact same prime factors as T(1,k)".
> >
> > So for example, Column 6 is the set whose members have prime factors
> > {2,5}: {10, 20, 40, 50, 80, 100, 160, 200...}.
> >
> > Is there a standard name for such sets?
> >
> > Cheers,
> > Bob Selcoe
> >
>
>
> ------------------------------
>
> Message: 16
> Date: Sat, 25 Mar 2017 17:31:59 -0500
> From: "Bob Selcoe" <rselcoe at entouchonline.net>
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Name for a type of set needed
> Message-ID: <D80249BA26BA48228706D062676E7CA0 at OwnerPC>
> Content-Type: text/plain; format=flowed; charset="utf-8";
>         reply-type=original
>
> Hi Max,
>
> > If you look at https://oeis.org/A284311/table you notice that there's an
> > error
> > in the data, leading to a value of 10 instead of 64.
>
> Thanks for catching that.  I missed 7 in the data and it threw things off.
>
> > IMHO it would be more natural to list the numbers with given prime
> factors
> > as rows...
>
> Perhaps, but at this point since it's published, I don't know if it's
> appropriate to change it.  Would you suggest we still do so?
>
> >I don't know a name for the set of numbers which has a given squarefree
> > kernel (A007947), one could simply say it's the inverse image
> > A007947^(-1) of that squarefree number.
>
> I was thinking of a more generic name that possibly could be used outside
> of
> OEIS, something *like* "squarefree kernel sets" (though perhaps there's a
> better name) because it seems to me it's a quite useful concept that could
> be applied in many ways - including analyzing "Yellowstone" type sequences
> (e.g., A098550, A064413, A280864).  It seems like a standard name does not
> exist.
>
> Cheers,
> Bob
>
> --------------------------------------------------
> From: "M. F. Hasler" <oeis at hasler.fr>
> Sent: Saturday, March 25, 2017 5:09 PM
> To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
> Subject: [seqfan] Re: Name for a type of set needed
>
> > If you look at https://oeis.org/A284311/table you notice that there's an
> > error
> > in the data, leading to a value of 10 instead of 64.
> >
> > IMHO it would be more natural to list the numbers with given prime
> factors
> > as rows, i.e., the transposed table, s.t. the first *column* would be
> > A005117 <https://oeis.org/A005117>,
> > in analogy with the similar oeis.org/A182944 <https://oeis.org/A182944>
> .
> >
> > but one could simply say it's the inverse image
> > A007947^(-1) of that squarefree number.
> >
> > Maximilian
> >
> > On Fri, Mar 24, 2017 at 9:24 PM, Bob Selcoe <rselcoe at entouchonline.net>
> > wrote:
> >
> >> Hi Seqfans,
> >>
> >> I just proposed sequence A284311, titled: "Array T(n,k) read by
> >> antidiagonals: T(1,k) = squarefree numbers > 1 in ascending order
> >> (A005117(k)); columns are non-squarefree numbers (in ascending order)
> >> with
> >> the exact same prime factors as T(1,k)".
> >>
> >> So for example, Column 6 is the set whose members have prime factors
> >> {2,5}: {10, 20, 40, 50, 80, 100, 160, 200...}.
> >>
> >> Is there a standard name for such sets?
> >>
> >> Cheers,
> >> Bob Selcoe
> >>
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
>
>