[math-fun] Sum of last ten digits

Wed Dec 28 03:05:21 CET 2005

Hi all,

Concerning Loops of Angelini's algorithm "Sum of last s digits" in base 10,
for s=7 Michaël Beeler found that ALL 9,999,999 beginning states fall in the
SAME loop (lenght 461).

> Michele Dondi wrote:
it may be interesting to investigate whether there are equivalent ones for
other bases and then a mathematically interesting sequence may be found.

Indeed, that was the idea when Mike Beeler computed this (posted the 9th
december).

---------------------
(All of these start with s-1 zeros followed by a one.)

     s=2 s=3 s=4 s=5 s=6 s=7  s=8   s=9  s=10
b=2    3   3   1   3   2   1    1     1     5
b=3    4   4   3   7   5   4   14    16    20
b=4    3   3   4  42  13  36    1     5    58
b=5    5   5   1  43   4  46    5    10    34
b=6    3  12  15 110  31 154  406     5   197
b=7    4   9   6  34  13  33   26  1440   104
b=8    6   6  45  84  41 249  171  6458  2801
b=9    9  30  14  52  16  74   20 14654    24
b=10   8   4  50 171  14 461   78 12203   312
b=11   4  14  30  10  36 332  666 16294  4686
b=12   5  10  26 116  39 603  120  6750 16105
b=13   7   9  61  57   9 263  130 13536   312
b=14   8  30  41  70  83 466   84 20008 16578
b=15  11  15   8 209  31 249 1010 31320 16806
b=16   9  12  54 224  25 666  312 19107 26294

         s=11    s=12     s=13    s=14     s=15     s=16
b=2         6       5        7       7        5        6
b=3         1       1       11       6        6        8
b=4         5       6        7     161        8       70
b=5         6      12      374      12      409        7
b=6       101     396        7     937      311      968
b=7        36     249      753     235     1478      794
b=8       687      88      676      35     3129     2533
b=9      2211      28       28      32       32     1093
b=10   318880    2184    57725    5804     1401     9722
b=11   220110    1456  3666650   99291        8    69188
b=12     9076  354312   576360  161050  1044670    26256
b=13   188448     728 23487352    1456 21972045      720
b=14  1541063    9520   226440   10248 55702156 62748516
b=15  1396913    2394  3281355 1098056 10309491    13680
b=16  1013143    1456    28640  852852 24039416   976560

(Thanks to Bill Gosper for the loop detector in HAKMEM item 132.)

A new large tail-to-loop ratio, for s=15, b=11, is
33668/8 = 4208.5.  The loop is 99, 99, 99, 99, 104,
99, 99, a3 (base 11; a=ten).

The loop lengths that occur above look rather random,
getting sparser as they get larger, except for 5 values:

loop length =  171 for s,b =  5,10 and  8,8
loop length =  249 for s,b =  7,8  and  7,15 and 12,7
loop length =  312 for s,b =  8,16 and 10,10 and 10,13
loop length =  666 for s,b =  7,16 and  8,11
loop length = 1456 for s,b = 12,11 and 12,16 and 14,13

Are these replications mere coincidence?  Or is there a simple
mapping of the structure between loops of the same length?

-- Mike

--------------------------

> Michele Dondi wrote:
> Since you write "first term" in quotes yourself, you must be aware that
> almost by definition, in a loop there's not a "first term".

Yes, my quotes did replace the fuzziness of two points of view taken
together: for one unique loop with one entry, <first term> makes sense; for
many entries, not anymore! although each of them is a <first term of the
loop>, from the point of view of each starting point! My short cut of
langage (for any common term between the cycle and a path leading to it)
should have better been replaced by <entry>.

> Personally I think that the entry point into the loop may not be that relevant
after all.

Who knows! It's a supposition. At this stage of the exploration, I would
check the entries before to conclude about their relevancy (relevancy in
terms of identity or of distribution along the loop): the unique cycle (with
s=7) doesn't have boundaries but does its basin of attraction have valleys
forcing trajectories from different starting points to converge and enter
the cycle through specific entries?

And what is really the system we try to render visible after all?! What is
it's nature regarding the concept of number? A starting point is not a point
but a succession of s digits: < a b c d e f g...>, < ab cd ef g...>, < ab
cdefg...> etc, i.e. all the 2^(s-1) compositions (ordered partitions) of s
digits. Any entry in our loop s=7 is linked to at least 64 different strings
of numbers/digits, or a multiple of it...

Does it makes sense to reduce this variability inherent in the algorithm,
connecting a <string of numbers> to <numbers>? How?!

[I tried s=? as a variable linked to the lenght of the beginning number: "s
= sum of last m digits", with m = the number of digits of the beginning
number. But it leads to some strange situation, I don't know what to think
about it:

Numbers from 0 to 9 are obviously fixed points.
But they are also the second step of a lot of longer numbers (of the form
ijk... with i+j+k+... < = 9 ) to which ad hoc algorithms have to be applied,
leading, them, to different loops! [cf. 2nd Table]. So one digit numbers are
at the same time fixed points and not!
To be AND not to be is that a question? Let's ask to quantum physicists!]

Michaël's 2nd table (cf. a preceding mail 17 dec):

  [base 10, nine starting points, s=2 to 16]
        s=2 s=3 s=4 s=5 s=6 s=7 s=8   s=9 s=10
digit=1   8   4  50 171  14 461  78 12203  312
digit=2   8  10  50 171  39 461  78 12203  312
digit=3   8  10  12 171  13 461  26 12203  104
digit=4   8   4  50 171  39 461  78 12203  312
digit=5   8   4  50  63  39 461  78 12203  312
digit=6   8  10  12 171  69 461  26 12203  104
digit=7   3  15  50 171  69 461  78 12203  312
digit=8   8   4  50 171  39 461  78 12203  312
digit=9   8  10   1 171   1 461   1 12203    1

           s=11  s=12   s=13  s=14  s=15  s=16
digit=1  318880  2184  57725  5804  1401  9722
digit=2  318880  2184  57725  5804  1401  3251
digit=3  318880   728  57725  5804  6698  9722
digit=4  318880  2184  57725  5804  1401  3251
digit=5  318880  2184  57725  5804  6698  9722
digit=6  318880   728  57725  5804  6698  9722
digit=7  318880  2184  57725  5804  6698  5091
digit=8  318880  2184  57725  5804  1401  9722
digit=9  318880     1  57725     1  6698     1

Anyway, what are the boundaries between the basins of attraction of these
loops? And what is their nature? Does the sets of the chains-of-numbers of
the boundaries (if any) have fractal properties (as usual in chaotic
systems)? Are there some numbers leading nowhere?

> Have you considered trying the opposite path? That is, you _do_ what the
> loop is like, don't you? Well, then for all of its entries try to devise
> the inverse image, i.e. where they do come from: the previous entry of the
> loop must be in it, of course;

Are you sure it's so evident?

> is it its only member? At least for one
> element of the loop this must not be the case, but it all boils down to:
> for how many of them this is not the case?

I think the answer is none, since all cases s=7 have been tested, and all
are leading into the loop (but I'm not shure I understand well your point).

Thanx for your interest and suggestions. I transmit to Michaël Beeler and
friends.

Alexandre

--------------------------------
"Sometimes I think the surest sign that intelligent life exists
elsewhere in the universe is that none of it has tried to contact us."
-- Bill  Watterson.

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