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

[Robert Gerbicz]

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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