[seqfan] Re: Fixpoints of the powertrain map (A133500) in different bases?

Thu Oct 3 20:19:51 CEST 2019

In English it is better to say "fixed point". "Fixpoint" is like a
fingernail scraping a blackboard.

Here are my notes about the calculations for the powertrain project:

$ cat README

(start)
This directory SEQS3 has the programs for the "powertrain" project with JHC
Dec 2007

subroutines has the basic programs

powertrain computes powertrain
zerotrain return 0 if powertrain(n) would be 0, otherwise return
powertrain(n)
PTtrajectory Compute trajectory of n under repeated application of the
powertrain map of A133500Returns -1 if the trajectory does not converge to
a single number in 100 steps
fasttraj =  Fast version Just returns length of trajectory or -1 if don't
hit single point after 100 steps
maxtraj returns high-water mark in trajecory

jobPT1  compute records for powertrain (one step) function Ran to 10^5. OK
jobPT2  compute records for length of trajectory. Ran out to 10^7.
        - rerunning Dec 21 2007 to check for nonconverging numbers to 10^7
jobPT4  compute records for high-water mark of trajectory - OK
jobPT5  stupid direct search for fixed points of powertrain
jobPT6  search 2^i 3^j 5^k 7^l for diverging, fixed, record length, range
ML to M, to 10^50 OK
jobPT7  search 2^i 3^j 5^k 7^l for biggest 2nd term, but not interesting
jobPT10 search 2^i 3^j 5^k 7^l for max number with nonzero powertrain,
fixed pts, to 10^100
        - used wrong zerotrain, but immaterial since have bigger values
from jobPT8 etc
jobPT11 faster version of jobPT10
jobPT8  looks for one solution & stops searches 3^j 5^k 7^l in the range
10^(L-5) to 10^L
jobPT8all looks for max solution searches all 3^j 5^k 7^l in the range
10^(L-5) to 10^L
jobPT8_5 looks for one solution & stops searches 2^i 3^j 7^l in the range
10^(L-5) to 10^L

jobPT12  tries to find a big list of n'' numbers

"Powerback" sequences:
jobPT13  searches for fixed points of powerback
jobPT14  computes time to reach loop

powertrain function:  A133500
powertrain function: A133500(n)<n: A133506, A133500(n)>n: A133507
powertrain function: fixed points: A135385
powertrain function: records: A133504, A133505
powertrain function: length of trajectory of n: A133501, A133502
powertrain function: records for length of trajectory: A133503, A133508
powertrain function: high point in trajectory of n: A135381; records:
A135382
powertrain function: numbers that converge to 2592: A135384

powerback function:  A133048
powerback function: fixed points: A131571
powerback function: records: A133059, A133134

(end)

Best regards
Neil

Neil J. A. Sloane, President, OEIS Foundation.
11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
Phone: 732 828 6098; home page: http://NeilSloane.com
Email: njasloane at gmail.com

On Thu, Oct 3, 2019 at 1:43 PM Georg.Fischer <georg.fischer at t-online.de>
wrote:

> The recent article in the German "Spektrum"
> magazine led me to <https://oeis.org/A135385>
> "Fixed points of the map m -> powertrain(m)" (2007),
> and I was astonished and puzzled by the terms
> 2592, 24547284284866560000000000 and the comments:
> - Probably there are no other terms.
> - There are no other terms below 10^100.
>
> In an attempt to get some additional idea I used a
> brute-force program which calculates the powertrain map
> (A133500) not only in decimal, but *in different bases*.
> It's not fast, so I only ran it up to n <= 30000.
> It shows the following non-trivial (> base) fixpoints:
>
> - none for base 2, 3, 4, 5
> - 16(10)    -> ... 16(10)=24(6)
> - none for 7
> - 27(10)    -> ... 27(10)=33(8)
>    230(10)   -> ... 486(10)=746(8)->14406(10)=34106(8)->486(10)=746(8)
>    3196(10)  -> ... 14406(10)=34106(8)->486(10)=746(8)
>                     ->14406(10)=34106(8)
>    [base 8 leads to a *cycle with two elements*; checked up to n=500000]
> - 5344(10)  -> ... 24586240(10)=51232874(9)
> - 129(10)   -> ... 7996018508417728512(10)=372b9a830000000000(12)
>    486(10)   -> ... 486(10)=346(12)
>    509(10)   -> ... 39366(10)=1a946(12)
>    1082(10)  -> ... 29282(10)=14b42(12)
>    9895(10)  -> ... 819200000000(10)=11292450a0a8(12)
> - none for 13, 14, 15
> - 25143(10) -> ... 78732(10)=1338c(16)
> - 29652(10) -> ... 10000(10)=20a4(17)
> - 24679(10) -> ... 768(10)=26c(18)
> - 1604(10)  -> ... 524288(10)=40862(19)
> - 124(10)   -> ... 1296(10)=34g(20)
> - 937(10)   -> ... 1024(10)=26g(21)
> - 27082(10) -> ... 2048(10)=452(22)
> - none for 23
> - 4116(10)  -> ... 4116(10)=73c(24)
> - none for 25
> - 85(10)    -> ... 2187(10)=363(26)
> - none for 27, 28
>
> Now my questions:
> - Are such fixpoint lists (especially for base 8, 12)
>    worth to become OEIS sequences?
> - Should I try to write a faster program, and
>    examine higher ranges?
> - Obviously a much more efficient program or argument
>    was used for the check <= 10^100 mentioned in A135385?
>    Which one?
>
> Regards - Georg
>
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>