[seqfan] Re: How fast are A034090, A034091 growing?

[Hugo Pfoertner]

On closer inspection, the ratio A034091(n)/A034091(n) is by no means a
constant, but increases approximately logarithmically with n.  You can see
this in this diagram
https://1drv.ms/u/s!ApCditH_KIhmgqY8qP8bOD8WbsH4Kw?e=4EsGXW
which was generated from the following file, which combines the two b-files
and has the quotient as the last column.
https://1drv.ms/t/s!ApCditH_KIhmgqY9gzCEkEIZBmNuZQ?e=hILHLw

On Mon, Jan 16, 2023 at 4:04 PM jean-paul allouche <
jean-paul.allouche at imj-prg.fr> wrote:

> Hi
>
> After having had a quick look through the relevant literature (where
> "record numbers"
> are also called "champion numbers"), I asked Jean-Louis Nicolas who is
> in particular
> a specialist of this kind of questions. He responded that, to his
> knowledge, all results
> about champion numbers for a function f make the assumption that f is
> multiplicative
> (recall that f is multiplicative if and only if f(mn) = f(m) f(n) as
> soon as gcd(m,n) = 1).
> The function here (sigma(n) - n) is not multiplicative, so that it is
> probably a quite
> difficult question to study its champion numbers. By the way J.-L.
> Nicolas told me that
> the word "champion" comes from Erdös, with whom he often spoke about
> these questions.
>
> best
> jean-paul
>
>
> Le 16/01/2023 à 04:46, Neil Sloane a écrit :
> > Actually I was hoping that someone would fit a curve to the 2750 terms
> that
> > are known for each of those two sequences
> > Best regards
> > Neil
> >
> > Neil J. A. Sloane, Chairman, OEIS Foundation.
> > Also Visiting Scientist, Math. Dept., Rutgers University,
> > Email: njasloane at gmail.com
> >
> >
> >
> > On Sun, Jan 15, 2023 at 6:51 PM Frank Adams-watters via SeqFan <
> > seqfan at list.seqfan.eu> wrote:
> >
> >> If you take a close look at the plot, you will see that the apparent
> ratio
> >> is about 5, not 1. Look at the labels on the axes.
> >>
> >> Franklin T. Adams-Watters
> >>
> >>
> >> -----Original Message-----
> >> From: Robert Gerbicz <robert.gerbicz at gmail.com>
> >> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> >> Sent: Sun, Jan 15, 2023 1:30 pm
> >> Subject: [seqfan] Re: How fast are A034090, A034091 growing?
> >>
> >> For A034091 see https://mathworld.wolfram.com/GronwallsTheorem.html,
> this
> >> is not exactly what you want.
> >>
> >> For A034090 the related highly abundant number of terms is known,
> wikipedia
> >> https://en.wikipedia.org/wiki/Highly_abundant_number says that there
> are
> >> log(N)^2 numbers up to N, a result of Erdos, not checked the article,
> >> pretty long.
> >>
> >> Neil Sloane <njasloane at gmail.com> ezt írta (időpont: 2023. jan. 15., V,
> >> 19:53):
> >>
> >>> Hugo,  about the ratio of those two sequences, all I did was use the
> >> Plot 2
> >>> button (at the bottom of any OEIS page), to plot A034090 vs A034091,
> and
> >>> the result is astonishingly close to a dead straight tine (with
> >> wobbles). I
> >>> guess the slope isn't 1, maybe it will turn out to be something like
> log
> >>> log n.
> >>>
> >>> Best regards
> >>> Neil
> >>>
> >>> Neil J. A. Sloane, Chairman, OEIS Foundation.
> >>> Also Visiting Scientist, Math. Dept., Rutgers University,
> >>> Email: njasloane at gmail.com
> >>>
> >>>
> >>>
> >>> On Sun, Jan 15, 2023 at 1:45 PM <hv at crypt.org> wrote:
> >>>
> >>>> I don't have answers, but the ratio seems the more likely to have been
> >>>> studied in the past.
> >>>>
> >>>> :Also their ratio seems close to 1, can that be made more precise?
> >>>>
> >>>> Not sure what you mean here - the last values in the b-files give a
> >> ratio
> >>>> around 5.35, and that will continue to grow.
> >>>>
> >>>> This is probably just random, but I note that if we call the ratio
> r_i,
> >>>> then the r_2750'th root of A034090(2750) is close to r_2750#.
> >>>>
> >>>> (1034758594602532800 ^ (1 / 5.3525611989641) ~= 2320.85710246872)
> >>>>
> >>>> Hugo
> >>>>
> >>>> Neil Sloane <njasloane at gmail.com> wrote:
> >>>> :Let f(n) = sigma(n)-n, the sum of the divisors d of n with d < n.
> >>>> :The n for which f(n) reaches a new record high, and the corresponding
> >>>> :values of f(n), are A034090 and A034091.
> >>>> :
> >>>> :The question is, how fast are these growing?
> >>>> :
> >>>> :Also their ratio seems close to 1, can that be made more precise?
> >>>> :
> >>>> :I'm not hoping for anything rigorous.  I would be happy with a
> >>>> :statistician's estimate.
> >>>>
> >>>> --
> >>>> Seqfan Mailing list - http://list.seqfan.eu/
> >>>>
> >>> --
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