[seqfan] Re: A program to compute A002845

[Vladimir Reshetnikov]

A friend of mine, Kirill Osenkov, used my program and a computer with 432
GB RAM to find the next two elements of the sequence, it took about 7
hours; they are A002845(28) = 688,821,573 and A002845(29) = 1,647,853,491.
He estimated the computation of the next term would require about 4TB RAM,
but we are trying to come up with some ideas how to use less memory. Based
on several extrapolation approaches, I expect that A002845(30) ≈
3,948,160,000 ± 300,000.

Does anybody have an idea about asymptotic behavior of this sequence? The
known values can be closely approximated by A002845(n) ≈ 0.34708262 *
2.518831816^n * (log(n) / n)^2.0941666.

--
Best regards
Vladimir Reshetnikov


On Thu, Jan 31, 2019 at 2:26 AM bacher <Roland.Bacher at univ-grenoble-alpes.fr>
wrote:

>
> Nice. Thank you very much,
>
> Best wishes, Roland Bacher
>
> Vladimir Reshetnikov <v.reshetnikov at gmail.com> a écrit :
>
> > Yes, I believe there are.
> >
> > For example, the equality (((x^x)^x)^x)^x = ((x^x)^x)^(x^x) holds for x =
> > 2, but does not hold for x = -0.0802980827... + 4.4082663977... i,
> although
> > it satisfies (x^x)^x = x^(x^x).
> >
> > --
> > Best regards
> > Vladimir Reshetnikov
> >
> >
> > On Wed, Jan 30, 2019 at 6:06 AM bacher <
> Roland.Bacher at univ-grenoble-alpes.fr>
> > wrote:
> >
> >>
> >> Are there any identities not coming from
> >> ((2^2)^2)=(2^(2^2)) among such expressions?
> >>
> >> Best wishes, Roland Bacher
> >>
> >>
> >> Vladimir Reshetnikov <v.reshetnikov at gmail.com> a écrit :
> >>
> >> > Today I published a program on GitHub that computes elements of
> >> > https://oeis.org/A002845 (Number of distinct values taken by
> 2^2^...^2,
> >> > with n 2's and parentheses inserted in all possible ways).
> >> >
> >> > https://github.com/VladimirReshetnikov/Oeis.A002845
> >> >
> >> > It uses a special representation of numbers that allows to efficiently
> >> > manipulate high power towers, but is still brute force in the sense
> that
> >> it
> >> > enumerates and stores all possible outcomes for each n. So far I was
> able
> >> > to confirm all values currently listed (up to n=27), and I estimate
> that
> >> I
> >> > would need a machine with more than 100GB RAM to compute the next.
> >> >
> >> > I would be delighted if anybody could improve my solution, port it to
> >> other
> >> > programming languages, or provide computational resources.
> >> >
> >> > --
> >> > Best regards
> >> > Vladimir Reshetnikov
> >> >
> >> > --
> >> > Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >>
> >>
> >> --
> >> Seqfan Mailing list - http://list.seqfan.eu/
> >>
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
>
>
>
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