Comma Sums

(plus a tree by Aai)

J’ai envoyé ceci à la liste SeqFan le 4 octobre 2011 sous le titre (ampoulé) « First differences are the sums of the two digits touching the commas » :

> Hello SeqFans,

>

> S = 0,5,11,13,17,26,35,44,53,62,71,79,97,105,111,113,117,...

>

> Consider the first comma; its closest digits are 0 and 5;

> add 0 to 5; the sum (5) is the first difference between 0 and 5.

>

> Consider the second comma; its closest digits are 5 and 1;

> add 5 to 1; the sum (6) is the first difference between 5 and 11.

> Etc.

>

> Does S stop at some point? If yes, could another -longer- S arise from a different start?

>

> Best,

> É.

Jack Brennen répondit quelques laps plus tard :

This sequence stops after 123 terms. The 123rd term is 989 which has no possible successor.

It looks like many low starting points lead to 989 and halt there.

The smallest starting point which goes beyond that seems to be the number 396 (which has no possible predecessor).

The sequence beginning with 396 goes on for quite some distance.

I traced it up to values in excess of 20 million (over 3 million terms).

[Éric] to Jack Brennen:

Thank you, Jack -- this is exactly what I was looking for,

Best,

É.

(some integers can be followed by two integers, as you will have noticed:

14 --> 19 or 20

28 --> 39 or 40

... but this doesn’t change much)

Lars Blomberg:

Hello Eric,

As for 396, I notice that the first differences of the sequence are:

396,10,10,10,10,10,10,10,10,10,10,11,12,14,8,6,12,14,8,6,12,15,

10,10,10,10,10,10,10,10,11,12,14,8,16,12,14,8,17,14,8,16,12,14,

8,16,13,16,12,14,18,16,12,(6,2,4,8)

... repeated indefinitely, which suggests that the sequence is infinite.

This looks like an interesting class of sequences (with a simple formulation!)

More investigations will follow, if you are interested.

Regards,

Lars

-----------

(...)

Hello again,

I immediately discovered that the first differences starting from 0,5 are

5,6,2,4,9,9,9,9,9,9,8,18,8,

6,2,4,8,

6,2,4,8,

6,2,4,8,

6,2,4,8,

6,2,4,9,8,

6,2,4,8,

6,2,4,8,

6,2,4,8,

6,2,4,8,

7,4,8,

6,12,4,8,

6,12,4,8,

6,12,4,9,8,

6,12,4,8,

6,12,4,8,

6,12,4,9,8,

6,12,14,8,6,12,14,8,6,13,6,12,14,8,6,12,14,8,6,13,16,12,14,8,16,12,15,

10,10,10,10,10,10,10,10,10,11,12,14,18,16,12,14

So repetitions of 6,2,4,8 may occur and still give a finite sequence.

More to investigate...

----------

(...)

Hello Eric,

Iterating 396 for a while I reached the number 101,709,259,615 after 12,490,000,000 iterations.

Further:

2.1    Numbers with no successors

Empirically for number of digits n=2,3,4:

86    986    9986

87    987    9987

88    988    9988

89    989    9989

91    991    9991

92    992    9992

93    993    9993

94    994    9994

It seems that the rule is that numbers of the form:

10^n - s, s=14, 13, 12, 11, 9, 8, 7, 6

have no successors. Can this be proven? It suggests that once an iteration has come into the millions, the "chance" of hitting a no-successor becomes very small.

2.2    Numbers with 2 successors

The first few numbers with 2 successors.

9    19    20

33    39    40

47    59    60

71    79    80

189    199    200

393    399    400

587    599    600

791    799    800

1989    1999    2000

3993    3999    4000

5987    5999    6000

7991    7999    8000

19989    19999    20000

39993    39999    40000

59987    59999    60000

79991    79999    80000

199989    199999    200000

399993    399999    400000

599987    599999    600000

799991    799999    800000

1999989    1999999    2000000

3999993    3999999    4000000

5999987    5999999    6000000

7999991    7999999    8000000

19999989    19999999    20000000

39999993    39999999    40000000

59999987    59999999    60000000

79999991    79999999    80000000

The rule seems to be that one of the successors start with an odd digit followed by all 9’s, the other starts with the next even digit followed by all 0’s. Can this be proven? From this it follows that numbers with 2 successors are of the form:

Predecessor    Successor 1    Successor 2

2*10^n-11      1*10^n-1       2*10^n

4*10^n-7       3*10^n-1       4*10^n

6*10^n-13      5*10^n-1       6*10^n

8*10^n-9       7*10^n-1       8*10^n

(No number seems to have more than 3 successors. Can this be proven?)

Continuing...

/Lars

Aai (private mail):

Hi Eric,

here are some more numbers with 2 successors. The pattern is obvious I think (if not already known by you) :-)

9    14    28    33    47    52    66    71

189   194   388   393   587   592   786   791

1989  1994  3988  3993  5987  5992  7986  7991

19989 19994 39988 39993 59987 59992 79986 79991

This computes the two versions of the series starting with 0, 5:

use warnings;

use strict;

sub succ {

my (\$x) = @_;

my \$end = \$x % 10;

my @ok;

return map {

my \$nextx = \$x + \$end + \$_;

(\$nextx =~ /^\$_/) ? (\$nextx) : ()

} (1 .. 9);

}

my \$i = 0;

print "[", ++\$i, "] 0\n";

for (my @curr = (5); @curr; @curr = map { succ(\$_) } @curr) {

print "[", ++\$i, "] @curr\n";

}

The results are:

[1] 0

[2] 5

[3] 11

[4] 13

[5] 17

[6] 26

[7] 35

[8] 44

[9] 53

[10] 62

[11] 71

[12] 79 80

[13] 97 88

[14] 105

[15] 111

[16] 113

[17] 117

[18] 125

[19] 131

[20] 133

[21] 137

[22] 145

[23] 151

[24] 153

[25] 157

[26] 165

[27] 171

[28] 173

[29] 177

[30] 185

[31] 191

[32] 193

[33] 197

[34] 206

[35] 214

[36] 220

[37] 222

[38] 226

[39] 234

[40] 240

[41] 242

[42] 246

[43] 254

[44] 260

[45] 262

[46] 266

[47] 274

[48] 280

[49] 282

[50] 286

[51] 294

[52] 301

[53] 305

[54] 313

[55] 319

[56] 331

[57] 335

[58] 343

[59] 349

[60] 361

[61] 365

[62] 373

[63] 379

[64] 391

[65] 395

[66] 404

[67] 412

[68] 418

[69] 430

[70] 434

[71] 442

[72] 448

[73] 460

[74] 464

[75] 472

[76] 478

[77] 490

[78] 494

[79] 503

[80] 511

[81] 517

[82] 529

[83] 543

[84] 551

[85] 557

[86] 569

[87] 583

[88] 591

[89] 597

[90] 610

[91] 616

[92] 628

[93] 642

[94] 650

[95] 656

[96] 668

[97] 682

[98] 690

[99] 696

[100] 709

[101] 725

[102] 737

[103] 751

[104] 759

[105] 775

[106] 787

[107] 802

[108] 812

[109] 822

[110] 832

[111] 842

[112] 852

[113] 862

[114] 872

[115] 882

[116] 892

[117] 903

[118] 915

[119] 929

[120] 947

[121] 963

[122] 975

[123] 989

To compute all possible versions of the series starting from 0, change the last part to:

my \$i = 0;

for (my @curr = (0); @curr; @curr = map { succ(\$_) } @curr) {

print "[", ++\$i, "] @curr\n";

}

And you still get 123 lines, ending with:

[114] 872 882 882 892 892 882

[115] 882 892 892 903 903 892

[116] 892 903 903 915 915 903

[117] 903 915 915 929 929 915

[118] 915 929 929 947 947 929

[119] 929 947 947 963 963 947

[120] 947 963 963 975 975 963

[121] 963 975 975 989 989 975

[122] 975 989 989 989

[123] 989

The program was written for ease of coding, not for speed -- I wouldn’t’ve used Perl if I was going for speed. But, changing (0) to (396), I find that in 2 minutes it computes about 1/4 million terms. I’ll start it running now and come back later and see what happens.

(...)

I killed it when it had used 1 hour of CPU time. The sequence began:

[1] 396

[2] 406

[3] 416

[4] 426

[5] 436

[6] 446

[7] 456

[8] 466

[9] 476

[10] 486

[11] 496

[12] 507

[13] 519

[14] 533

[15] 541

[16] 547

[17] 559

[18] 573

[19] 581

[20] 587

[21] 599 600

The two versions eventually came back together with a 1-step offset:

[250] 1981 1973

[251] 1983 1977

[252] 1987 1985

[253] 1995 1991

[254] 2002 1993

[255] 2006 1997

[256] 2014 2006

That is, 2,006 was the 255th or 256th term depending on which choice was taken for the 21st term.

Because I hadn’t programmed to check for that situation, it went on computing both versions of the series until I killed it. There were no further bifurcations and the last value computed and written out was 52,531,251, which was the 8,134,537th or 8,134,538th term. I was curious enough about this to write a C version of the program, designed to terminate if the series bifurcated again, ended, or exceeded a value of 1 billion. I checked it on the starting values 5 and 396 and got the results already seen.

Then I started it on a value of 2006:

[1] 2006

[2] 2014

[3] 2020

[4] 2022

[5] 2026

[6] 2034

[7] 2040

[8] 2042

[9] 2046

[10] 2054

[11] 2060

[12] 2062

[13] 2066

[14] 2074

[15] 2080

[16] 2082

[17] 2086

[18] 2094

[19] 2100

[20] 2102

[21] 2106

[22] 2114

[23] 2120

and several hours and about 2 gigabytes of disk space later, it terminated due to passing 1 billion:

[121481135] 999999667

[121481136] 999999683

[121481137] 999999695

[121481138] 999999709

[121481139] 999999727

[121481140] 999999743

[121481141] 999999755

[121481142] 999999769

[121481143] 999999787

[121481144] 999999803

[121481145] 999999815

[121481146] 999999829

[121481147] 999999847

[121481148] 999999863

[121481149] 999999875

[121481150] 999999889

[121481151] 999999907

[121481152] 999999923

[121481153] 999999935

[121481154] 999999949

[121481155] 999999967

[121481156] 999999983

[121481157] 999999995

So, starting from 396 as the first term, we will reach 1,000,000,001 as the 121,481,412th or 121,481,413th term, and the series is still going, with no more bifurcations after the one at the 21st term.

Here’s the program:

#include <stdio.h>

#include <stdlib.h>

int main (int c, char **v)

{

long i, n, succ (long);

if (c != 2) {

fprintf (stderr, "Usage\n");

return 1;

}

n = atol (v[1]);

for (i = 1; n >= 0; ++i, n = succ (n))

printf ("[%d] %d\n", i, n);

return 0;

}

long succ (long n)

{

long n_plus_end = n + n % 10;

int dig, start;

long next1 = -1, next2;

int dig1 (long);

for (dig = 1; dig < 10; ++dig) {

next2 = n_plus_end + dig;

start = dig1 (next2);

if (dig == start) {

if (next1 > 0) {

printf ("Bifurcation: %ld -> %ld or %ld\n",

n, next1, next2);

return -1;

}

next1 = next2;

}

}

if (next1 < 0) printf ("End\n");

return next1;

}

int dig1 (long n)

{

/* I think this should be faster than doing / in a loop */

if (n < 10000) {

if (n < 100) {

if (n < 10) return n;

else return n/10;

} else {

if (n < 1000) return n/100;

else return n/1000;

}

} else if (n < 100000000) {

if (n < 1000000) {

if (n < 100000) return n/10000;

else return n/100000;

} else {

if (n < 10000000) return n/1000000;

else return n/10000000;

}

} else if (n < 1000000000) return n/100000000;

else {

printf ("Cannot handle n = %ld\n", n);

exit (1);

}

}

Nicolas Graner:

I am surprised that you haven’t noticed that the sequence starting with 396 is infinite.

Starting from 1001 (which is reached from 396), you eventually reach 9995. The next number is 10001, which eventually yields 99995, followed by 100001. Notice a pattern? :)

I haven’t written a formal proof, but it should be pretty easy, by induction over the number of digits. Here is the idea:

Take 1x001, where x is any sequence of digits. It will reproduce the sequence from 1001 to 1995, up to 1x995. This is followed by 1y001, with y = X+1. This repeats until x becomes all 9’s, at which point the first digit will change: 199..995 -> 200..002.

Now we get numbers of the form 2x002, which reproduce the sequence from 2002 to 2994 while x increases, until it reaches all 9’s and the first digit becomes 3.

You can repeat this reasoning with each value of the first digit. There are some subtleties because, when the first digit changes, the last one is not always the same. But since the sequence is very "robust" (as evidenced by forking sequences, which reunite pretty quickly), variations in the last digit have no effect beyond a couple of terms.

So, starting with 100*001, we are bound to get eventually to 999*995, which is followed with 1000*001 (with one more digit). Which shows that the sequence is infinite.

OK, mathematically speaking I can’t claim the sequence is infinite until someone works out every detail of the proof, but I’d be really surprised it the "obvious" pattern didn’t repeat indefinitely.

Nicolas

Jack Brennen:

I figured that it probably was infinite, but proving it requires a whole lot more time than stating it.  :)

An outline of a proof would be to prove that each of the following nine relationships is valid, where z represents some number of consecutive zeroes (possibly including no zeroes -- the empty string):

Jean-Marc Falcoz:

Salut Eric,

Inutile de te dire que j’aime bien cette suite ! Je n’ai pas tenu compte des cas un nombre a deux successeurs possibles, j’ai pris le plus petit qui marche. Voici donc deux ou trois résultats pêle-mêle :

1) le graphe de la suite commençant par 396 (un peu plus de 200000 termes).

Comme souvent avec les suites "presque" périodiques, un graphe illustre bien la situation : on voit ces espèces de chaînettes, vers 12000, 120000, et 1200000 (je suppose)

2) Une liste de nombre sans successeurs :

{1, 2, 3, 4, 86, 87, 88, 89, 91, 92, 93, 94, 986, 987, 988, 989, 991, 992, 993, 994, 9987, 9989, 9991, 9993}

3) une liste partielle (1 à 2000, puis 8000 à 9000) avec le premier terme, le dernier (si la suite s’arrête, sinon le premier terme dépassant 1000000), et le nombre de termes (surtout utile si la suite s’arrête)

{{1, 1, 1}, {2, 2, 1}, {3, 3, 1}, {4, 4, 1}, {5, 989, 122}, {6, 989, 121}, {7, 989, 121}, {8, 989, 120}, {9, 989, 120}, {10, 989, 122}, {11, 989, 121}, {12, 989, 121}, {13, 989, 120}, {14, 989, 120}, {15, 989, 120}, {16, 989, 120}, {17, 989, 119}, {18, 989, 118}, {19, 989, 119}, {20, 989, 120}, {21, 989, 120}, {22, 989, 119}, {23, 989, 118}, {24, 989, 119}, {25, 989, 118}, {26, 989, 118}, {27, 989, 117}, {28, 989, 117}, {29, 989, 117}, {30, 989, 118}, {31, 989, 118}, {32, 989, 117}, {33, 989, 117}, {34, 989, 117}, {35, 989, 117}, {36, 989, 116}, {37, 989, 116}, {38, 989, 116}, {39, 989, 116}, {40, 989, 117}, {41, 989, 116}, {42, 989, 116}, {43, 989, 116}, {44, 989, 116}, {45, 989, 115}, {46, 989, 115}, {47, 88, 4}, {48, 989, 115}, {49, 88, 4}, {50, 989, 115}, {51,  989, 115}, {52, 88, 4}, {53, 989, 115}, {54, 88, 4}, {55, 989, 114}, {56, 989, 114}, {57, 989, 114}, {58, 989, 114}, {59, 88, 3}, {60, 989, 114}, {61, 989, 114}, {62, 989, 114}, {63, 989, 114}, {64,  88, 3}, {65, 93, 3}, {66, 989, 113}, {67, 93, 3}, {68, 989, 113}, {69, 86, 2}, {70, 93, 3}, {71, 989, 113}, {72, 93, 3}, {73, 989, 113}, {74, 86, 2}, {75, 88, 2}, {76, 91, 2}, {77, 93, 2}, {78, 989, 113}, {79, 989, 112}, {80, 88, 2}, {81, 91, 2}, {82, 93, 2}, {83, 989, 113}, {84, 989, 112}, {85, 989, 102}, {86, 86, 1}, {87, 87, 1}, {88, 88, 1}, {89, 89, 1}, {90, 989, 102}, {91, 91, 1}, {92, 92, 1}, {93, 93, 1}, {94, 94, 1}, {95, 989, 112}, {96, 989,  111}, {97, 989, 111}, {98, 989, 110}, {99, 989, 101}, {100, 989, 112}, {101, 989, 111}, {102, 989, 111}, {103, 989, 110}, {104, 989, 101}, {105, 989, 110}, {106, 989, 109}, {107, 989, 109}, {108, 989, 108}, {109, 989, 100}, {110, 989, 110}, {111, 989, 109}, {112, 989, 109}, {113, 989, 108}, {114, 989, 100}, {115, 989, 108}, {116, 989, 107}, {117, 989, 107}, {118, 989, 106}, {119, 989, 99}, {120, 989, 108}, {121, 989, 107}, {122, 989, 107}, {123, 989, 106}, {124, 989, 99}, {125, 989, 106}, {126, 989, 105}, {127, 989, 105}, {128, 989, 104}, {129, 989, 98}, {130, 989, 106}, {131, 989, 105}, {132, 989, 105}, {133, 989, 104}, {134, 989, 98}, {135, 989, 104}, {136, 989, 103}, {137, 989, 103}, {138, 989, 102}, {139, 989, 97}, {140, 989, 104}, {141, 989, 103}, {142, 989, 103}, {143, 989, 102}, {144, 989, 97}, {145, 989, 102}, {146, 989, 101}, {147, 989, 101}, {148, 989, 100}, {149, 989, 96}, {150, 989, 102}, {151, 989, 101}, {152, 989, 101}, {153, 989, 100}, {154, 989, 96}, {155, 989, 100}, {156, 989, 99}, {157, 989, 99}, {158, 989, 98}, {159, 989, 95}, {160, 989, 100}, {161, 989, 99}, {162, 989, 99}, {163, 989, 98}, {164, 989, 95}, {165, 989, 98}, {166, 989, 97}, {167, 989, 97}, {168, 989, 96}, {169, 989, 94}, {170, 989, 98}, {171, 989, 97}, {172, 989, 97}, {173, 989, 96}, {174, 989, 94}, {175, 989, 96}, {176, 989, 95}, {177, 989, 95}, {178, 989, 94}, {179, 989, 93}, {180, 989, 96}, {181, 989, 95}, {182, 989, 95}, {183, 989, 94}, {184, 989, 93}, {185, 989, 94}, {186, 989, 93}, {187, 989, 93}, {188, 989, 92}, {189, 989, 92}, {190, 989, 94}, {191, 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__________

...

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120388}, {8334, 9991, 153}, {8335, 9989, 120}, {8336, 9991, 153}, {8337, 9989, 133}, {8338, 1000001, 120386}, {8339, 1000001, 120387}, {8340, 9989,  120}, {8341, 9991, 153}, {8342, 9989, 133}, {8343, 1000001, 120386}, {8344, 1000001, 120387}, {8345, 9991, 152}, {8346, 9991, 152}, {8347, 9989, 132}, {8348, 9989, 119}, {8349, 1000001, 120385}, {8350, 9991, 152}, {8351, 9991, 152}, {8352, 9989, 132}, {8353, 9989, 119}, {8354, 1000001, 120385}, {8355, 9991, 151}, {8356, 1000001, 120386}, {8357, 9989, 131}, {8358, 9991, 151}, {8359, 9989, 118}, {8360, 9991, 151}, {8361, 1000001, 120386}, {8362, 9989, 131}, {8363, 9991, 151}, {8364, 9989, 118}, {8365, 1000001, 120385}, {8366, 1000001, 120384}, {8367, 9989,  130}, {8368, 9991, 150}, {8369, 9991, 150}, {8370, 1000001, 120385}, {8371, 1000001, 120384}, {8372, 9989, 130}, {8373, 9991, 150}, {8374, 9991, 150}, {8375, 1000001, 120383}, {8376, 9989, 117}, {8377, 9989, 129}, {8378, 1000001, 120384}, {8379, 9991, 149}, {8380, 1000001, 120383}, {8381, 9989, 117}, {8382, 9989, 129}, {8383, 1000001, 120384}, {8384, 9991, 149}, {8385, 9989, 116}, {8386, 9991, 149}, {8387, 9989, 128}, {8388, 1000001, 120382}, {8389, 1000001, 120383}, {8390, 9989, 116}, {8391, 9991, 149}, {8392, 9989, 128}, {8393, 1000001, 120382}, {8394, 1000001, 120383}, {8395, 9991, 148}, {8396, 9991, 148}, {8397, 9989, 127}, {8398, 9989, 115}, {8399, 1000001, 120381}, {8400, 9991, 148}, {8401, 9991, 148}, {8402, 9989, 127}, {8403, 9989, 115}, {8404, 1000001, 120381}, {8405, 9991, 147}, {8406, 1000001, 120382}, {8407, 9989, 126}, {8408, 9991, 147}, {8409, 9989, 114}, {8410, 9991, 147}, {8411, 1000001, 120382}, {8412, 9989, 126}, {8413, 9991, 147}, {8414, 9989, 114}, {8415, 1000001, 120381}, {8416, 1000001, 120380}, {8417, 9989, 125}, {8418, 9991, 146}, {8419, 9991, 146}, {8420, 1000001, 120381}, {8421, 1000001, 120380}, {8422, 9989, 125}, {8423, 9991, 146}, {8424, 9991, 146}, {8425, 1000001, 120379}, {8426, 9989, 113}, {8427, 9989, 124}, {8428, 1000001, 120380}, {8429, 9991, 145}, {8430, 1000001, 120379}, {8431, 9989, 113}, {8432, 9989, 124}, {8433, 1000001, 120380}, {8434, 9991, 145}, {8435, 9989, 112}, {8436, 9991, 145}, {8437, 9989, 123}, {8438, 1000001, 120378}, {8439, 1000001, 120379}, {8440, 9989, 112}, {8441, 9991, 145}, {8442, 9989, 123}, {8443, 1000001, 120378}, {8444, 1000001, 120379}, {8445, 9991,  144}, {8446, 9991, 144}, {8447, 9989, 122}, {8448, 9989, 111}, {8449, 1000001, 120377}, {8450, 9991, 144}, {8451, 9991, 144}, {8452, 9989, 122}, {8453, 9989, 111}, {8454, 1000001, 120377}, {8455, 9991, 143}, {8456, 1000001, 120378}, {8457, 9989, 121}, {8458, 9991, 143}, {8459, 9989, 110}, {8460, 9991, 143}, {8461, 1000001, 120378}, {8462, 9989, 121}, {8463, 9991, 143}, {8464, 9989, 110}, {8465, 1000001, 120377}, {8466, 1000001, 120376}, {8467, 9989, 120}, {8468, 9991, 142}, {8469, 9991, 142}, {8470, 1000001, 120377}, {8471, 1000001, 120376}, {8472, 9989,  120}, {8473, 9991, 142}, {8474, 9991, 142}, {8475, 1000001, 120375}, {8476, 9989, 109}, {8477, 9989, 119}, {8478, 1000001, 120376}, {8479, 9991, 141}, {8480, 1000001, 120375}, {8481, 9989, 109}, {8482, 9989, 119}, {8483, 1000001, 120376}, {8484, 9991, 141}, {8485, 9989, 108}, {8486, 9991, 141}, {8487, 9989, 118}, {8488, 1000001, 120374}, {8489, 1000001, 120375}, {8490, 9989,  108}, {8491, 9991, 141}, {8492, 9989, 118}, {8493, 1000001, 120374}, {8494, 1000001, 120375}, {8495, 9991, 140}, {8496, 9991, 140}, {8497, 9989, 117}, {8498, 9989, 107}, {8499, 1000001, 120373}, {8500, 9991, 140}, {8501, 9991, 140}, {8502, 9989, 117}, {8503, 9989, 107}, {8504, 1000001, 120373}, {8505, 9991, 139}, {8506, 1000001, 120374}, {8507, 9989, 116}, {8508, 9991, 139}, {8509, 9989, 106}, {8510, 9991, 139}, {8511, 1000001, 120374}, {8512, 9989, 116}, {8513, 9991, 139}, {8514, 9989, 106}, {8515, 1000001, 120373}, {8516, 1000001, 120372}, {8517, 9989,  115}, {8518, 9991, 138}, {8519, 9991, 138}, {8520, 1000001, 120373}, {8521, 1000001, 120372}, {8522, 9989, 115}, {8523, 9991, 138}, {8524, 9991, 138}, {8525, 1000001, 120371}, {8526, 9989, 105}, {8527, 9989, 114}, {8528, 1000001, 120372}, {8529, 9991, 137}, {8530, 1000001, 120371}, {8531, 9989, 105}, {8532, 9989, 114}, {8533, 1000001, 120372}, {8534, 9991, 137}, {8535, 9989, 104}, {8536, 9991, 137}, {8537, 9989, 113}, {8538, 1000001, 120370}, {8539, 1000001, 120371}, {8540, 9989, 104}, {8541, 9991, 137}, {8542, 9989, 113}, {8543, 1000001, 120370}, {8544, 1000001, 120371}, {8545, 9991, 136}, {8546, 9991, 136}, {8547, 9989, 112}, {8548, 9989, 103}, {8549, 1000001, 120369}, {8550, 9991, 136}, {8551, 9991, 136}, {8552, 9989, 112}, {8553, 9989, 103}, {8554, 1000001, 120369}, {8555, 9991, 135}, {8556, 1000001, 120370}, {8557, 9989, 111}, {8558, 9991, 135}, {8559, 9989, 102}, {8560, 9991, 135}, {8561, 1000001, 120370}, {8562, 9989, 111}, {8563, 9991, 135}, {8564, 9989, 102}, {8565, 1000001, 120369}, {8566, 1000001, 120368}, {8567, 9989, 110}, {8568, 9991, 134}, {8569, 9991, 134}, {8570, 1000001, 120369}, {8571, 1000001, 120368}, {8572, 9989, 110}, {8573, 9991, 134}, {8574, 9991, 134}, {8575, 1000001, 120367}, {8576, 9989, 101}, {8577, 9989, 109}, {8578, 1000001, 120368}, {8579, 9991, 133}, {8580, 1000001, 120367}, {8581, 9989, 101}, {8582, 9989, 109}, {8583, 1000001, 120368}, {8584, 9991, 133}, {8585, 9989, 100}, {8586, 9991, 133}, {8587, 9989, 108}, {8588, 1000001, 120366}, {8589, 1000001, 120367}, {8590, 9989, 100}, {8591, 9991, 133}, {8592, 9989, 108}, {8593, 1000001, 120366}, {8594, 1000001, 120367}, {8595, 9991,  132}, {8596, 9991, 132}, {8597, 9989, 107}, {8598, 9989, 99}, {8599, 1000001, 120365}, {8600, 9991, 132}, {8601, 9991, 132}, {8602, 9989, 107}, {8603, 9989, 99}, {8604, 1000001, 120365}, {8605, 9991, 131}, {8606, 1000001, 120366}, {8607, 9989, 106}, {8608, 9991, 131}, {8609, 9989, 98}, {8610, 9991, 131}, {8611,  1000001, 120366}, {8612, 9989, 106}, {8613, 9991, 131}, {8614, 9989, 98}, {8615, 1000001, 120365}, {8616, 1000001, 120364}, {8617, 9989, 105}, {8618, 9991, 130}, {8619, 9991, 130}, {8620, 1000001, 120365}, {8621, 1000001, 120364}, {8622, 9989, 105}, {8623, 9991, 130}, {8624, 9991, 130}, {8625, 1000001, 120363}, {8626, 9989, 97}, {8627, 9989, 104}, {8628, 1000001, 120364}, {8629, 9991, 129}, {8630, 1000001, 120363}, {8631, 9989, 97}, {8632, 9989, 104}, {8633, 1000001, 120364}, {8634, 9991, 129}, {8635, 9989, 96}, {8636, 9991, 129}, {8637, 9989, 103}, {8638, 1000001, 120362}, {8639,  1000001, 120363}, {8640, 9989, 96}, {8641, 9991, 129}, {8642, 9989,  103}, {8643, 1000001, 120362}, {8644, 1000001, 120363}, {8645, 9991, 128}, {8646, 9991, 128}, {8647, 9989, 102}, {8648, 9989, 95}, {8649, 1000001, 120361}, {8650, 9991, 128}, {8651, 9991, 128}, {8652, 9989, 102}, {8653, 9989, 95}, {8654, 1000001, 120361}, {8655, 9991, 127}, {8656, 1000001, 120362}, {8657, 9989, 101}, {8658, 9991, 127}, {8659, 9989, 94}, {8660, 9991, 127}, {8661,  1000001, 120362}, {8662, 9989, 101}, {8663, 9991, 127}, {8664, 9989, 94}, {8665, 1000001, 120361}, {8666, 1000001, 120360}, {8667, 9989, 100}, {8668, 9991, 126}, {8669, 9991, 126}, {8670, 1000001, 120361}, {8671, 1000001, 120360}, {8672, 9989, 100}, {8673, 9991, 126}, {8674, 9991, 126}, {8675, 1000001, 120359}, {8676, 9989, 93}, {8677, 9989, 99}, {8678, 1000001, 120360}, {8679, 9991, 125}, {8680, 1000001, 120359}, {8681, 9989, 93}, {8682, 9989, 99}, {8683, 1000001, 120360}, {8684, 9991, 125}, {8685, 9989, 92}, {8686, 9991, 125}, {8687, 9989, 98}, {8688, 1000001, 120358}, {8689, 1000001, 120359}, {8690, 9989, 92}, {8691, 9991, 125}, {8692, 9989, 98}, {8693, 1000001, 120358}, {8694, 1000001, 120359}, {8695, 9991, 124}, {8696, 9991, 124}, {8697, 9989, 97}, {8698, 9989, 91}, {8699, 1000001, 120357}, {8700, 9991, 124}, {8701, 9991, 124}, {8702, 9989, 97}, {8703, 9989, 91}, {8704, 1000001, 120357}, {8705, 9991, 123}, {8706, 1000001, 120358}, {8707,  9989, 96}, {8708, 9991, 123}, {8709, 9989, 90}, {8710, 9991, 123}, {8711, 1000001, 120358}, {8712, 9989, 96}, {8713, 9991, 123}, {8714, 9989, 90}, {8715, 1000001, 120357}, {8716, 1000001, 120356}, {8717, 9989, 95}, {8718, 9991, 122}, {8719, 9991, 122}, {8720, 1000001, 120357}, {8721, 1000001, 120356}, {8722, 9989,  95}, {8723, 9991, 122}, {8724, 9991, 122}, {8725, 1000001, 120355}, {8726, 9989, 89}, {8727, 9989, 94}, {8728, 1000001, 120356}, {8729, 9991, 121}, {8730, 1000001, 120355}, {8731, 9989, 89}, {8732, 9989, 94}, {8733, 1000001, 120356}, {8734, 9991, 121}, {8735, 9989, 88}, {8736, 9991, 121}, {8737, 9989, 93}, {8738, 1000001, 120354}, {8739, 1000001, 120355}, {8740, 9989, 88}, {8741, 9991, 121}, {8742, 9989, 93}, {8743, 1000001, 120354}, {8744, 1000001, 120355}, {8745, 9991, 120}, {8746, 9991, 120}, {8747, 9989,  92}, {8748, 9989, 87}, {8749, 1000001, 120353}, {8750, 9991, 120}, {8751, 9991, 120}, {8752, 9989, 92}, {8753, 9989, 87}, {8754, 1000001, 120353}, {8755, 9991, 119}, {8756, 1000001, 120354}, {8757,  9989, 91}, {8758, 9991, 119}, {8759, 9989, 86}, {8760, 9991, 119}, {8761, 1000001, 120354}, {8762, 9989, 91}, {8763, 9991, 119}, {8764, 9989, 86}, {8765, 1000001, 120353}, {8766, 1000001, 120352}, {8767, 9989, 90}, {8768, 9991, 118}, {8769, 9991, 118}, {8770, 1000001, 120353}, {8771, 1000001, 120352}, {8772, 9989,  90}, {8773, 9991, 118}, {8774, 9991, 118}, {8775, 1000001, 120351}, {8776, 9989, 85}, {8777, 9989, 89}, {8778, 1000001, 120352}, {8779, 9991, 117}, {8780, 1000001, 120351}, {8781, 9989, 85}, {8782, 9989, 89}, {8783, 1000001, 120352}, {8784, 9991, 117}, {8785, 9989, 84}, {8786, 9991, 117}, {8787, 9989, 88}, {8788, 1000001, 120350}, {8789, 1000001, 120351}, {8790, 9989, 84}, {8791, 9991, 117}, {8792, 9989, 88}, {8793, 1000001, 120350}, {8794, 1000001, 120351}, {8795, 9991, 116}, {8796, 9991, 116}, {8797, 9989,  87}, {8798, 9989, 83}, {8799, 1000001, 120349}, {8800, 9991, 116}, {8801, 9991, 116}, {8802, 9989, 87}, {8803, 9989, 83}, {8804, 1000001, 120349}, {8805, 9991, 115}, {8806, 1000001, 120350}, {8807,  9989, 86}, {8808, 9991, 115}, {8809, 9989, 82}, {8810, 9991, 115}, {8811, 1000001, 120350}, {8812, 9989, 86}, {8813, 9991, 115}, {8814, 9989, 82}, {8815, 1000001, 120349}, {8816, 1000001, 120348}, {8817, 9989, 85}, {8818, 9991, 114}, {8819, 9991, 114}, {8820, 1000001, 120349}, {8821, 1000001, 120348}, {8822, 9989,  85}, {8823, 9991, 114}, {8824, 9991, 114}, {8825, 1000001, 120347}, {8826, 9989, 81}, {8827, 9989, 84}, {8828, 1000001, 120348}, {8829, 9991, 113}, {8830, 1000001, 120347}, {8831, 9989, 81}, {8832, 9989, 84}, {8833, 1000001, 120348}, {8834, 9991, 113}, {8835, 9989, 80}, {8836, 9991, 113}, {8837, 9989, 83}, {8838, 1000001, 120346}, {8839, 1000001, 120347}, {8840, 9989, 80}, {8841, 9991, 113}, {8842, 9989, 83}, {8843, 1000001, 120346}, {8844, 1000001, 120347}, {8845, 9991, 112}, {8846, 9991, 112}, {8847, 9989,  82}, {8848, 9989, 79}, {8849, 1000001, 120345}, {8850, 9991, 112}, {8851, 9991, 112}, {8852, 9989, 82}, {8853, 9989, 79}, {8854, 1000001, 120345}, {8855, 9991, 111}, {8856, 1000001, 120346}, {8857,  9989, 81}, {8858, 9991, 111}, {8859, 9989, 78}, {8860, 9991, 111}, {8861, 1000001, 120346}, {8862, 9989, 81}, {8863, 9991, 111}, {8864, 9989, 78}, {8865, 1000001, 120345}, {8866, 1000001, 120344}, {8867, 9989, 80}, {8868, 9991, 110}, {8869, 9991, 110}, {8870, 1000001, 120345}, {8871, 1000001, 120344}, {8872, 9989,  80}, {8873, 9991, 110}, {8874, 9991, 110}, {8875, 1000001, 120343}, {8876, 9989, 77}, {8877, 9989, 79}, {8878, 1000001, 120344}, {8879, 9991, 109}, {8880, 1000001, 120343}, {8881, 9989, 77}, {8882, 9989, 79}, {8883, 1000001, 120344}, {8884, 9991, 109}, {8885, 9989, 76}, {8886, 9991, 109}, {8887, 9989, 78}, {8888, 1000001, 120342}, {8889, 1000001, 120343}, {8890, 9989, 76}, {8891, 9991, 109}, {8892, 9989, 78}, {8893, 1000001, 120342}, {8894, 1000001, 120343}, {8895, 9991, 108}, {8896, 9991, 108}, {8897, 9989,  77}, {8898, 9989, 75}, {8899, 1000001, 120341}, {8900, 9991, 108}, {8901, 9991, 108}, {8902, 9989, 77}, {8903, 9989, 75}, {8904, 1000001, 120341}, {8905, 9991, 107}, {8906, 1000001, 120342}, {8907,  9989, 76}, {8908, 9991, 107}, {8909, 9989, 74}, {8910, 9991, 107}, {8911, 1000001, 120342}, {8912, 9989, 76}, {8913, 9991, 107}, {8914, 9989, 74}, {8915, 1000001, 120341}, {8916, 1000001, 120340}, {8917, 9989, 75}, {8918, 9991, 106}, {8919, 9991, 106}, {8920, 1000001, 120341}, {8921, 1000001, 120340}, {8922, 9989,  75}, {8923, 9991, 106}, {8924, 9991, 106}, {8925, 1000001, 120339}, {8926, 9989, 73}, {8927, 9989, 74}, {8928, 1000001, 120340}, {8929, 9991, 105}, {8930, 1000001, 120339}, {8931, 9989, 73}, {8932, 9989, 74}, {8933, 1000001, 120340}, {8934, 9991, 105}, {8935, 9989, 72}, {8936, 9991, 105}, {8937, 9989, 73}, {8938, 1000001, 120338}, {8939, 1000001, 120339}, {8940, 9989, 72}, {8941, 9991, 105}, {8942, 9989, 73}, {8943, 1000001, 120338}, {8944, 1000001, 120339}, {8945, 9991, 104}, {8946, 9991, 104}, {8947, 9989,  72}, {8948, 9989, 71}, {8949, 1000001, 120337}, {8950, 9991, 104}, {8951, 9991, 104}, {8952, 9989, 72}, {8953, 9989, 71}, {8954, 1000001, 120337}, {8955, 9991, 103}, {8956, 1000001, 120338}, {8957,  9989, 71}, {8958, 9991, 103}, {8959, 9989, 70}, {8960, 9991, 103}, {8961, 1000001, 120338}, {8962, 9989, 71}, {8963, 9991, 103}, {8964, 9989, 70}, {8965, 1000001, 120337}, {8966, 1000001, 120336}, {8967, 9989, 70}, {8968, 9991, 102}, {8969, 9991, 102}, {8970, 1000001, 120337}, {8971, 1000001, 120336}, {8972, 9989,  70}, {8973, 9991, 102}, {8974, 9991, 102}, {8975, 1000001, 120335}, {8976, 9989, 69}, {8977, 9989, 69}, {8978, 1000001, 120336}, {8979, 9991, 101}, {8980, 1000001, 120335}, {8981, 9989, 69}, {8982, 9989, 69}, {8983, 1000001, 120336}, {8984, 9991, 101}, {8985, 9989, 68}, {8986, 9991, 101}, {8987, 9989, 68}, {8988, 1000001, 120334}, {8989, 1000001, 120335}, {8990, 9989, 68}, {8991, 9991, 101}, {8992, 9989, 68}, {8993, 1000001, 120334}, {8994, 1000001, 120335}, {8995, 9987, 67}, {8996, 9991, 100}, {8997, 1000001, 119334}, {8998, 9989, 67}, {8999, 9993, 67}, {9000, 9987, 67}, {9001, 9991, 100}, {9002, 1000001, 119334}, {9003, 9989, 67}, {9004, 9993, 67}, {9005, 1000001, 120333}, {9006, 9991, 99}, {9007, 1000001, 120334}, {9008, 1000001, 119333}, {9009, 9987, 66}, {9010, 1000001, 120333}, {9011, 9991, 99}, {9012, 1000001, 120334}, {9013, 1000001, 119333}, {9014, 9987, 66}, {9015, 9989, 66}, {9016, 9991, 98}, {9017, 9993, 66}, {9018, 1000001, 120333}}

Aai – a complement:

Maybe the following is of some interest too.

If I’m not wrong the terminator values are (well-) known through the following patterns:

86 87 88 89 91 92 93 94

986 987 988 989 991 992 993 994

9986 9987 9988 9989 9991 9992 9993 9994

etc.

The next row is made by prepending a 9 to numbers of the previous row.

By working with predecessors we are able to build a complete tree of each terminator value. Each branch of such a tree is then one of Eric’s S sequences, e.g. take terminator value 987:

┌───────────────────────────┐

│987 974

├───────────────────────────┤

│987 969 960

├───────────────────────────┤

│987 969 955 943 932

├───────────────────────────┤

│987 969 955 943 927 914

├───────────────────────────┤

│987 969 955 943 927 909 900│

├───────────────────────────┤

│987 969 955 943 927 909 895│

├───────────────────────────┤

│987 969 955 938

└───────────────────────────┘

Count of ‘all’ branches of terminator 989: 330

Some end-leaves are counted more than once, e.g. 14 (has 2 successors):

19 31 ___

14  /          \ 35 44 53 62 71 79 97 105 111 113 117 125 131 -->  989

\ 20 22 26 /

But 14 has a smaller terminator than 989, because:

19 31 ___                 / 80 88

14 /          \ 35 44 53 62 71

\ 20 22 26 /                \ 79 97 105 111 113 117 125 131 --> 989

Unique starting values that end with 989 are:

5 6 7 8 9 10 12 14 16 18 21 23 25 27 29 30 32 34

36 38 41 43 45 47 50 52 56 58 61 63 78 83 85 90 96 98

100 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134

136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166 168 170

172 174 176 178 180 182 184 186 188 190 192 194 196 198 201 203 205 207

209 211 213 215 217 219 221 223 225 227 229 231 233 235 237 239 241 243

245 247 249 251 253 255 257 259 261 263 265 267 269 271 273 275 277 279

281 283 285 287 289 291 293 295 297 299 300 302 304 306 308 310 312 314

316 318 320 322 324 326 328 330 332 334 336 338 340 342 344 346 348 350

352 354 356 358 360 362 364 366 368 370 372 374 376 378 380 382 384 386

388 390 392 394 407 413 415 419 423 425 427 429 437 443 445 449 453 455

457 459 467 473 475 479 483 485 487 489 499 504 506 508 516 522 532 534

538 544 546 548 556 562 572 574 578 584 586 588 598 603 605 607 615 621

631 633 637 643 645 647 655 661 671 673 677 683 685 687 697 702 706 714

724 730 740 742 746 752 756 764 774 780 790 792 798 803 807 809 817 827

831 835 837 847 853 857 859 867 877 881 885 887 920 934 952 958 980

Example 504

504 513 521 527 539 553 561

567 579 593 602 610 616 628

642 650 656 668 682 690 696

709 725 737 751 759 775 787

802 812 822 832 842 852 862

872 882 892 903 915 929 947

963 975 989

Be aware: all of this is based on the assumption that there are only certain terminator values. :-)

--

Met vriendelijke groet,

=@@i

Claudio Meller:

Hi Eric, I found those numbers with no predecessors :

1, 2, 3, 4, 5, 6, 7, 8, 9,

10, 12, 14, 16, 18, 21, 23, 25, 27, 29, 30, 32, 34, 36, 38, 41, 43,

45, 47, 49, 50, 52, 54, 56, 58, 61, 63, 65, 67, 69, 70, 72, 74, 76,

78, 81, 83, 85, 87, 89, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108,

110, 112, 114, 116, 118, 120, 122, 124, 126, 128, 130, 132, 134, 136,

138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164,

166, 168, 170, 172, 174, 176, 178, 180, 182, 184, 186, 188, 190, 192,

194, 196, 198, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221,

223, 225, 227, 229, 231, 233, 235, 237, 239, 241, 243, 245, 247, 249,

251, 253, 255, 257, 259, 261, 263, 265, 267, 269, 271, 273, 275, 277,

279, 281, 283, 285, 287, 289, 291, 293, 295, 297, 299, 300, 302, 304,

306, 308, 310, 312, 314, 316, 318, 320, 322, 324, 326, 328, 330, 332,

334, 336, 338, 340, 342, 344, 346, 348, 350, 352, 354, 356, 358, 360,

362, 364, 366, 368, 370, 372, 374, 376, 378, 380, 382, 384, 386, 388,

390, 392, 394, 396, 398, 401, 403, 405, 407, 409, 411, 413, 415, 417,

419, 421, 423, 425, 427, 429, 431, 433, 435, 437, 439, 441, 443, 445,

447, 449, 451, 453, 455, 457, 459, 461, 463, 465, 467, 469, 471, 473,

475, 477, 479, 481, 483, 485, 487, 489, 491, 493, 495, 497, 499, 500,

502, 504, 506, 508, 510, 512, 514, 516, 518, 520, 522, 524, 526, 528,

530, 532, 534, 536, 538, 540, 542, 544, 546, 548, 550, 552, 554, 556,

558, 560, 562, 564, 566, 568, 570, 572, 574, 576, 578, 580, 582, 584,

586, 588, 590, 592, 594, 596, 598, 601, 603, 605, 607, 609, 611, 613,

615, 617, 619, 621, 623, 625, 627, 629, 631, 633, 635, 637, 639, 641,

643, 645, 647, 649, 651, 653, 655, 657, 659, 661, 663, 665, 667, 669,

671, 673, 675, 677, 679, 681, 683, 685, 687, 689, 691, 693, 695, 697,

699, 700, 702, 704, 706, 708, 710, 712, 714, 716, 718, 720, 722, 724,

726, 728, 730, 732, 734, 736, 738, 740, 742, 744, 746, 748, 750, 752,

754, 756, 758, 760, 762, 764, 766, 768, 770, 772, 774, 776, 778, 780,

782, 784, 786, 788, 790, 792, 794, 796, 798, 801, 803, 805, 807, 809,

811, 813, 815, 817, 819, 821, 823, 825, 827, 829, 831, 833, 835, 837,

839, 841, 843, 845, 847, 849, 851, 853, 855, 857, 859, 861, 863, 865,

867, 869, 871, 873, 875, 877, 879, 881, 883, 885, 887, 889, 891, 893,

895, 897, 899, 900, 902, 904, 906, 908, 910, 912, 914, 916, 918, 920,

922, 924, 926, 928, 930, 932, 934, 936, 938, 940, 942, 944, 946, 948,

950, 952, 954, 956, 958, 960, 962, 964, 966, 968, 970, 972, 974, 976,

978, 980, 982, 984, 986, 988, 990, 992, 994, 996, 998, 1000, ...

... an integer beginning with an odd digit has no predecessor if it ends also with an even digit (and vice versa).

Aai se propose (en néerlandais et mathématiques), de démontrer que :

> (...) ca. half of the numbers of [1 .. N] have no predecessors recognized by the property that the sum of leading and trailing digit has to be odd (a bit restated but also mentioned by Claudio Meller). I hope that a mathematician will have a critical look at the following :-)

Hoofdstelling:

Van de verzameling getallen 1 t/m N hebben (1 + N - 9)/2 getallen geen voorganger.

Hulpstelling:

Elk getal waarvan de som van het eerste en het laatste cijfer oneven is heeft geen voorganger

n-1

Gegeven: G = Sigma a(i)*10^i  en a(n-1) + a(0) = 2*k + 1

0

Bewijs:

n-1

Stel voorganger P = Sigma b(i)*10^i  met P < G, dan is:

0

Bedenk dat van P   b(n-1) = 0 kan zijn.

n-1

G - P = a(n-1) + b(0) = Sigma (a(i)-b(i))*10^i  + a(0) - b(0)

1

n-1

Sigma (a(i)-b(i))*10^i - 2*b(0) = a(n-1) - a(0)

1

of druk beide zijden uit in even/oneven expressies

2*r = 2*s + 1   is niet waar

--> P is geen voorganger van G

q.e.d

Hieruit volgt tevens de hoofdstelling, ervan uitgaande dat 0 als opvolgers 1 t/m

9 heeft.

--

Met vriendelijke groet,

=@@i

Aai (later):

A consequence of the fact, that numbers with an odd sum of msd and lsd have no predecessors, is:

- only half of the ‘terminator’-numbers are real terminators. A number like 986 is an isolated number, no successors as well as no predecessors.

Aai today (October 19th, 2011, upon a search for “solitaire” numbers – integers with no predecessor and no successor):

In fact it’s not difficult to produce them (pen & paper). Look at the following matrix of (potential) root (or terminator) numbers:

86   87   88   89   91   92   93   94

986  987  988  989  991  992  993  994

9986 9987 9988 9989 9991 9992 9993 9994

From each row, 4 of them are root numbers and the other 4 are solitaires. Because the pattern of these numbers is obvious, extending the matrix is easy.

E.g. the 5-th row will be (adding 9’s at the beginning):

999986 999987 999988 999989 999991 999992 999993 999994

From these rows only numbers with an odd sum of msd and lsd are “solitaires”.

Here are the first few (are they really of some interest?)

87           89           92           94

986          988          992          994

9986         9988         9992         9994

99986        99988        99992        99994

999986       999988       999992       999994

9999986      9999988      9999992      9999994

99999986     99999988     99999992     99999994

999999986    999999988    999999992    999999994

9999999986   9999999988   9999999992   9999999994

99999999986  99999999988  99999999992  99999999994

999999999986 999999999988 999999999992 999999999994

...

I checked the missing numbers of the in the interval [0... 1000] and they are:

87 89 92 94 986 988 992 994

... as expected.

_________

[FRENCH] :

On a compris le procédé :

- deux nombres séparés par une virgule diffèrent d’une quantité égale à la somme des chiffres qui touchent la virgule;

- la plupart des « petits nombres » produisent des suites finies qui bloquent sur 989 (ou d’autres);

- 396 semble être le plus petit entier ne bloquant pas ;

- certain entiers n’ont pas de prédécesseur, d’autres pas de successeur, d’autres ni l’un ni l’autre (le dernier tableau ci-dessus, celui des « solitaires » proposé par Aai)

__________

A remark from Aai about his:

Also, it’s probably obvious, but the picture suggests that numbers that end in e.g. 1001 are also trees. That’s of course not true. The graph is about the interval [0 ... 999] and numbers > 999 are kind of picture overshoot and do not tell how they continue.

(...)

And this last message, also by Aai:

I think I have a more or less clear picture of this coupling of numbers:

I conjecture that all commasum sequences are or finite because they end in a root number (has no successors, e.g. 989) or are infinite because they start as branches (with one or more members) of an infinite tree of which the smallest source number (starts a commasum sequence) is equal to 396.

Per interval of 10^n .. 10^(n+1) - 1 you can distinguish 4 finite trees and 4 solitaires, all the other numbers are part of an infinite tree.

I propose that if a sequence has the choice of joining the ‘infinite’ seq. or a seq. that ends in a root, then the seq. is finite.

I checked the numbers of the graph that run out of the picture, e.g. starting with 996 --> 1003: they all join the sequence that starts with 396 at some point in the next interval.

But I still cannot proof that 396 runs forever.

--

Met vriendelijke groet,

=@@i

__________

Thank you to Jack, Lars, Aai, Mark, Jean-Marc, Nicolas Graner and Claudio for their comments & computations.

__________