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<TITLE>RE: [math-fun] Sum of last ten digits</TITLE>
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Hi all,<BR>
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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). <BR>
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<FONT COLOR="#0000FD">> Michele Dondi wrote: <BR>
it may be interesting to investigate whether there are equivalent ones for other bases and then a mathematically interesting sequence may be found.<BR>
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Indeed, that was the idea when Mike Beeler computed this (posted the 9th december).<BR>
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(All of these start with s-1 zeros followed by a one.)<BR>
<BR>
s=2 s=3 s=4 s=5 s=6 s=7 s=8 s=9 s=10<BR>
b=2 3 3 1 3 2 1 1 1 5<BR>
b=3 4 4 3 7 5 4 14 16 20<BR>
b=4 3 3 4 42 13 36 1 5 58<BR>
b=5 5 5 1 43 4 46 5 10 34<BR>
b=6 3 12 15 110 31 154 406 5 197<BR>
b=7 4 9 6 34 13 33 26 1440 104<BR>
b=8 6 6 45 84 41 249 171 6458 2801<BR>
b=9 9 30 14 52 16 74 20 14654 24<BR>
b=10 8 4 50 171 14 461 78 12203 312<BR>
b=11 4 14 30 10 36 332 666 16294 4686<BR>
b=12 5 10 26 116 39 603 120 6750 16105<BR>
b=13 7 9 61 57 9 263 130 13536 312<BR>
b=14 8 30 41 70 83 466 84 20008 16578<BR>
b=15 11 15 8 209 31 249 1010 31320 16806<BR>
b=16 9 12 54 224 25 666 312 19107 26294<BR>
<BR>
s=11 s=12 s=13 s=14 s=15 s=16<BR>
b=2 6 5 7 7 5 6<BR>
b=3 1 1 11 6 6 8<BR>
b=4 5 6 7 161 8 70<BR>
b=5 6 12 374 12 409 7<BR>
b=6 101 396 7 937 311 968<BR>
b=7 36 249 753 235 1478 794<BR>
b=8 687 88 676 35 3129 2533<BR>
b=9 2211 28 28 32 32 1093<BR>
b=10 318880 2184 57725 5804 1401 9722<BR>
b=11 220110 1456 3666650 99291 8 69188<BR>
b=12 9076 354312 576360 161050 1044670 26256<BR>
b=13 188448 728 23487352 1456 21972045 720<BR>
b=14 1541063 9520 226440 10248 55702156 62748516<BR>
b=15 1396913 2394 3281355 1098056 10309491 13680<BR>
b=16 1013143 1456 28640 852852 24039416 976560<BR>
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(Thanks to Bill Gosper for the loop detector in HAKMEM item 132.)<BR>
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A new large tail-to-loop ratio, for s=15, b=11, is<BR>
33668/8 = 4208.5. The loop is 99, 99, 99, 99, 104,<BR>
99, 99, a3 (base 11; a=ten).<BR>
<BR>
The loop lengths that occur above look rather random,<BR>
getting sparser as they get larger, except for 5 values:<BR>
<BR>
loop length = 171 for s,b = 5,10 and 8,8<BR>
loop length = 249 for s,b = 7,8 and 7,15 and 12,7<BR>
loop length = 312 for s,b = 8,16 and 10,10 and 10,13<BR>
loop length = 666 for s,b = 7,16 and 8,11<BR>
loop length = 1456 for s,b = 12,11 and 12,16 and 14,13<BR>
<BR>
Are these replications mere coincidence? Or is there a simple<BR>
mapping of the structure between loops of the same length?<BR>
<BR>
-- Mike<BR>
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<FONT COLOR="#0000FD">> Michele Dondi wrote:<BR>
> Since you write "first term" in quotes yourself, you must be aware that <BR>
> almost by definition, in a loop there's not a "first term". <BR>
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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>.<BR>
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<FONT COLOR="#0000FD">> Personally I think that the entry point into the loop may not be that relevant after all.<BR>
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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? <BR>
<BR>
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... <BR>
<BR>
<BR>
Does it makes sense to reduce this variability inherent in the algorithm, connecting a <string of numbers> to <numbers>? How?! <BR>
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[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:<BR>
<BR>
Numbers from 0 to 9 are obviously fixed points. <BR>
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! <BR>
To be AND not to be is that a question? Let's ask to quantum physicists!]<BR>
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Michaël's 2nd table (cf. a preceding mail 17 dec):<BR>
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[base 10, nine starting points, s=2 to 16] <BR>
<FONT COLOR="#7E0000"> s=2 s=3 s=4 s=5 s=6 s=7 s=8 s=9 s=10<BR>
digit=1 8 4 50 171 14 461 78 12203 312<BR>
digit=2 8 10 50 171 39 461 78 12203 312<BR>
digit=3 8 10 12 171 13 461 26 12203 104<BR>
digit=4 8 4 50 171 39 461 78 12203 312<BR>
digit=5 8 4 50 63 39 461 78 12203 312<BR>
digit=6 8 10 12 171 69 461 26 12203 104<BR>
digit=7 3 15 50 171 69 461 78 12203 312<BR>
digit=8 8 4 50 171 39 461 78 12203 312<BR>
digit=9 8 10 1 171 1 461 1 12203 1<BR>
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s=11 s=12 s=13 s=14 s=15 s=16<BR>
digit=1 318880 2184 57725 5804 1401 9722<BR>
digit=2 318880 2184 57725 5804 1401 3251<BR>
digit=3 318880 728 57725 5804 6698 9722<BR>
digit=4 318880 2184 57725 5804 1401 3251<BR>
digit=5 318880 2184 57725 5804 6698 9722<BR>
digit=6 318880 728 57725 5804 6698 9722<BR>
digit=7 318880 2184 57725 5804 6698 5091<BR>
digit=8 318880 2184 57725 5804 1401 9722<BR>
digit=9 318880 1 57725 1 6698 1<BR>
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<BR>
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?<BR>
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<FONT COLOR="#0000FD">> Have you considered trying the opposite path? That is, you _do_ what the <BR>
> loop is like, don't you? Well, then for all of its entries try to devise <BR>
> the inverse image, i.e. where they do come from: the previous entry of the <BR>
> loop must be in it, of course; <BR>
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Are you sure it's so evident?<BR>
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<FONT COLOR="#0000FD">> is it its only member? At least for one <BR>
> element of the loop this must not be the case, but it all boils down to: <BR>
> for how many of them this is not the case?<BR>
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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).<BR>
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Thanx for your interest and suggestions. I transmit to Michaël Beeler and friends.<BR>
<BR>
<BR>
Alexandre<BR>
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--------------------------------<BR>
"Sometimes I think the surest sign that intelligent life exists<BR>
elsewhere in the universe is that none of it has tried to contact us."<BR>
-- Bill Watterson.<BR>
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