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

[Hugo Pfoertner]

Fit of terms > 10^12 with Excel:
Linear: y = 5.3531x - 2E+15, R² = 0.9999,
Polynomial deg. 2: y = 9E-20x2 + 5.2898x - 9E+14, R² = 1

On Mon, Jan 16, 2023 at 4:47 AM Neil Sloane <njasloane at gmail.com> wrote:

> 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.
> > > >
> > > > --
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