[seqfan] Re: May 30 2021 Report

[Antti Karttunen]

On 5/30/21, Neil Sloane <njasloane at gmail.com> wrote:
> Dear SeqFans:
>
>
> 1. Gave a talk about OEIS to students who arrived at Rutgers for the summer
> program "Research Experience for Undergraduates" (REU). I was going to
> mention some recent sequences from the Drafts stack, for example :
>
> A343376 & A343456; A344570; A343369; A344538
>
> and I was going to say:
>
> - it is always worth looking at the"stack", https://oeis.org/drafts,
> because there are many interesting sequences there every day, and when you
> see something interesting you should think about questions like:
>
> -- is the definition clear? is the sequence well-defined?
>
> -- is the sequence infinite? is there a proof?
>
> -- how fast does it grow?
>
> -- does every number appear?
>
> -- is there a formula or generating function?
>
> -- how would you write a program to generate it?
>
> - If the sequence entry doesn't answer these questions, and you can, you
> should submit a comment to the entry (of course you should register).
> Getting your name into the OEIS is something to be proud of.
>
> It was a Zoom talk, so of course there were technical problems, and nothing
> went as planned. Most of it was recorded.

Great! Is this (yet) anywhere publicly or semi-publicly available?

> The sequences I mentioned are
> probably still worth investigating.
>
>
> 1a: To find other worthwhile problems to work on, in the OEIS search
> window, enter "It would be nice", or "Conjecture", or "Empirical".
>
>
> 1b. Or, start at a recent sequence, say A344555 (interesting sequence, by
> the way), and walk backwards through the recently accepted sequences. By
> that I mean, enter A344555 in the search window, then when it appears,
> click on its A-number. This will show the "Adjacent sequences" link at the
> bottom. Now you can walk backwards through the OEIS.
>
>
> 2. Ali Sada's recent question on this list led me to the almost-core
> sequence A011772, the smallest m such that m(m+1)/2 is divisible by n. It
> is interesting because it is a basic question and it does not have a simple
> answer. There is no simple formula for a(n). Look at the graph. I don't
> even have an adjective to describe it. Yet I've seen many other sequences
> with a similar graph. The Index to the OEIS has an entry for:
>
> Graphs (or plots) , sequences with interesting:
>
> that gives names ("Tornado", etc)
>
> but A011772 is not listed (as I said, there's no name for this graph).
>
>
> 3.  So I was thinking - and I was almost going to mention this to the
> students - us OEIS folks spend a lot of time staring at some sequence,
> trying to figure out what is going on.  Looking for any kind of formula,
> and ANYTHING that would help understand it (see example in next paragraph).
> It might be helpful if we had some way of grouping sequences according to
> their graphs. The EKG sequence for example is basically three straight
> lines of different slopes. The Yellowstone permutation is similar but has
> more lines. Superseeker might have a component that would tell you "Your
> sequence has a graph which is rather like the graphs of the following
> sequences ..."  If there as a close match that might be really helpful.

Yes, yes, yes! But please note that such "emanating rays" (rising sun
hiding in the origin) type of scatter plot is _really_ common, and
there must be thousands, or even myriads of such sequences in OEIS,
mostly appearing for many "basic number theoretical sequences" (for a
lack of more precise categorization). So, we would really need an "AI"
(more prosaically, an image-classifying algorithm) to sort out the
subtle differences between such sequences, e.g., based on the slopes
of the densest rays.


Now, for more easily detected differences of this pattern, here's a
variant where the rays appear in two distinct lobes of "fans":

https://oeis.org/A344440/graph

Its appearance is easy to explain, because A344440  is just A061020
rotated 45 degrees counterclockwise, and A061020's formula (Negate
primes in factorizations of divisors of n, then sum) explains its
divided appearance and the emptiness of the area around x-axis.

But here is another one with a similar appearance:
https://oeis.org/A342917/graph
defined as a(n) = A001615(n) / gcd(1+n, A001615(n)), where A001615 is
Dedekind psi, n * Product_{p|n, p prime} (1 + 1/p).

Now, for what reason the divided lobes? And if we rotate this
clockwise 45 degrees, then what would a(n) = n - A342917(n) mean?

BTW, I computed a few sequences derived from A011772. Both sigma and
A005187 seem to give an upper bound for it.


Best regards,

Antti

>
>
> 4. Quite often when working on a problem I come across a sequence with a
> simple definition that is easy to generate but has no formula, and
> Superseeker doesn't help.
>
> There are several examples in this paper that we recently finished
> revising:
>
> Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration
> and Stained Glass Windows, 1: Rectangular Grids, Preprint, revised May 21
> 2021, http://neilsloane.com/doc/rose_5.pdf,
>
>
> for example A334701, the number of simple interior nodes in the graph
> BC(1,n). We have 500 terms, but no formula!
>
> See Table 6 in the paper, or A334701.
>
> Following the example of one of my friends: if you can find a formula for
> A334701, I will donate $100 to the OEIS Foundation in your name. (See
> http://oeisf.org/#DONORS.)
>
>
> 5. Do we have any Mac experts here? I do all my work in xterm windows, and
> vim suddenly stopped working. I was unable to install a new version using
> Homebrew. I am running High Sierra 10.13.6. If you could help, send me an
> email.
>
>
> 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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