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

[Hugo Pfoertner]

Another fit with a power seems to indicate that the growth rate might be
slightly superlinear, i.e., A034091(n) ~= 3.4152 * A034090(n) ^ 1.0111.

On Mon, Jan 16, 2023 at 8:25 AM Hugo Pfoertner <yae9911 at gmail.com> wrote:

> 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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>> > >
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