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

[Neil Sloane]

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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> > >
> >
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>
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