Hello
MathFun,
[hope no mistakes here, calculated by hand -- and a first
warm thank you to Michael Kleber,
Franklin T. Adams-Watters and Joshua Zucker]
Say
we have a rule which transforms an integer in another one like this:
-
Add to n its odd digits and subtract
its even ones.
So
738 will become 740 (738+7+3-8)
103 will become 107 (103+1+3)
and 358 stays 358
(358+3+5-8).
Let’s
loop this procedure for integers 0 to 100 and see what happens:
0
-> 0
(a 1-loop or fixed point)
1 -> 2 -> 0 (1-loop)
2 -> 0 (1-loop)
3 -> 6 -> 0 (1-loop)
4 -> 0 (1-loop)
5 -> 10 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11 (a 6-loop)
6 -> 0 (1-loop)
7 -> 14 -> 11 -> 13 -> 17 -> 25 -> 28 -> 18 -> 11 (6-loop)
8 -> 0 (1-loop)
9 -> 18
-> 11 -> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)
10
-> 11 -> 13 -> 17 -> 25
-> 28 -> 18 -> 11 (6-loop)
11 -> 13 -> 17 -> 25
-> 28 -> 18 -> 11 (6-loop)
12
-> 11 -> 13 -> 17 -> 25
-> 28 -> 18 -> 11 (6-loop)
13 -> 17 -> 25 -> 28
-> 18 -> 11 -> 13 (6-loop)
14
-> 11 -> 13 -> 17 -> 25
-> 28 -> 18 -> 11 (6-loop)
15
-> 21 -> 20 -> 18 -> 11
-> 13 -> 17 -> 25 -> 28 -> 18 (6-loop)
16
-> 11 -> 13 -> 17 -> 25
-> 28 -> 18 -> 11 (6-loop)
17 -> 25 -> 28 -> 18
-> 11 -> 13 -> 17 (6-loop)
18 -> 11 -> 13 -> 17
-> 25 -> 28 -> 18 (6-loop)
19
-> 29 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 ->
77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 ->
130 -> 134 (1-loop or fixed point)
20
-> 18 -> 11 -> 13 -> 17
-> 25 -> 28 -> 18 (6-loop)
21
-> 20 -> 18 -> 11 -> 13
-> 17 -> 25 -> 28 -> 18 (6-loop)
22
-> 18 -> 11 -> 13 -> 17
-> 25 -> 28 -> 18 (6-loop)
23
-> 24 -> 18 -> 11 -> 13
-> 17 -> 25 -> 28 -> 18 (6-loop)
24
-> 18 -> 11 -> 13 -> 17
-> 25 -> 28 -> 18 (6-loop)
25 -> 28 -> 18 -> 11
-> 13 -> 17 -> 25 (6-loop)
26
-> 18 -> 11 -> 13 -> 17
-> 25 -> 28 -> 18 (6-loop)
27
-> 32 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 ->
91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
28 -> 18 -> 11 -> 13
-> 17 -> 25 -> 28 (6-loop)
29
-> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 ->
91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
30
-> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 ->
101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
31
-> 35 -> 43 -> 42 -> 36 -> 33 -> 39 -> 51 -> 57 ->
69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122
-> 119 -> 130 -> 134 (1-loop)
32
-> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 ->
101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
33
-> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 ->
103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)
34
-> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 ->
101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)
35
-> 43 -> 42 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 ->
72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 ->
119 -> 130 -> 134 (1-loop)
36
-> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 ->
101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
37
-> 47 -> 50 -> 55 -> 65
-> 64 -> 54 -> 55 (4-loop)
38
-> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 -> 91 ->
101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
39
-> 51 -> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 ->
107 -> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)
40
-> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 ->
91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
41
-> 38 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 ->
91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
42
-> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 ->
91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
43
-> 42 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 ->
77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 ->
130 -> 134
(1-loop)
44
-> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 ->
91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
45
-> 46 -> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 ->
77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 ->
130 -> 134
(1-loop)
46
-> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 ->
91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
47
-> 50 -> 55 -> 65 -> 64
-> 54 -> 55 (4-loop)
48
-> 36 -> 33 -> 39 -> 51 -> 57 -> 69 -> 72 -> 77 ->
91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
49
-> 54 -> 55 -> 65 -> 64
-> 54 -> 55 (4-loop)
50
-> 55 -> 65 -> 64 -> 54
-> 55 (4-loop)
51
-> 57 -> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107
-> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)
52
-> 55 -> 65 -> 64 -> 54
-> 55 (4-loop)
53
-> 61 -> 56 -> 55 -> 65
-> 64 -> 54 -> 55 (4-loop)
54 -> 55 -> 65 -> 64
-> 54 (4-loop)
55 -> 65 -> 64 -> 54
-> 55 (4-loop)
56
-> 55 -> 65 -> 64 -> 54
-> 55 (4-loop)
57
-> 69 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115
-> 122 -> 119 -> 130 -> 134 (1-loop)
58
-> 55 -> 65 -> 64 -> 54
-> 55 (4-loop)
59
-> 73 -> 83 -> 78 -> 77 -> 91 -> 101 -> 103 -> 107
-> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)
60
-> 54 -> 55 -> 65 -> 64
-> 54 (4-loop)
61
-> 56 -> 55 -> 65 -> 64
-> 54 -> 55 (4-loop)
62
-> 54 -> 55 -> 65 -> 64
-> 54 (4-loop)
63
-> 60 -> 54 -> 55 -> 65
-> 64 -> 54 (4-loop)
64 -> 54 -> 55 -> 65
-> 64 (4-loop)
65 -> 64 -> 54 -> 55
-> 65 (4-loop)
66
-> 54 -> 55 -> 65 -> 64
-> 54 (4-loop)
67
-> 68 -> 54 -> 55 -> 65
-> 64 -> 54 (4-loop)
68
-> 54 -> 55 -> 65 -> 64
-> 54 (4-loop)
69
-> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122
-> 119 -> 130 -> 134
(1-loop)
70
-> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 ->
130 -> 134
(1-loop)
71
-> 79 -> 95 -> 109 -> 119 -> 130 -> 134
(1-loop)
72
-> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119
-> 130 -> 134
(1-loop)
73
-> 83 -> 78 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115
-> 122 -> 119 -> 130 -> 134
(1-loop)
74
-> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119
-> 130 -> 134
(1-loop)
75
-> 87 -> 86 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107
-> 115 -> 122 -> 119 -> 130 -> 134 (1-loop)
76
-> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119
-> 130 -> 134
(1-loop)
77
-> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130
-> 134
(1-loop)
78
-> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119
-> 130 -> 134
(1-loop)
79
-> 95 -> 109 -> 119 -> 130 -> 134
(1-loop)
80
-> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122
-> 119 -> 130 -> 134
(1-loop)
81
-> 74 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122
-> 119 -> 130 -> 134
(1-loop)
82
-> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122
-> 119 -> 130 -> 134
(1-loop)
83
-> 78 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122
-> 119 -> 130 -> 134
(1-loop)
84
-> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122
-> 119 -> 130 -> 134
(1-loop)
85
-> 82 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115
-> 122 -> 119 -> 130 -> 134
(1-loop)
86
-> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122
-> 119 -> 130 -> 134
(1-loop)
87
-> 86 -> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115
-> 122 -> 119 -> 130 -> 134
(1-loop)
88
-> 72 -> 77 -> 91 -> 101 -> 103 -> 107 -> 115 -> 122
-> 119 -> 130 -> 134
(1-loop)
89
-> 90 -> 99 -> 117 -> 126 -> 119 -> 130 -> 134
(1-loop)
90
-> 99 -> 117 -> 126 -> 119 -> 130 -> 134
(1-loop)
91
-> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
92
-> 99 -> 117 -> 126 -> 119 -> 130 -> 134 (1-loop)
93
-> 105 -> 111 -> 114 -> 112
(1-loop)
94
-> 99 -> 117 -> 126 -> 119 -> 130 -> 134 (1-loop)
95
-> 108 -> 101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130
-> 134
(1-loop)
96
-> 99 -> 117 -> 126 -> 119 -> 130 -> 134 (1-loop)
97
-> 113 -> 118 -> 112
(1-loop)
98
-> 99 -> 117 -> 126 -> 119 -> 130 -> 134
(1-loop)
99
-> 117 -> 126 -> 119 -> 130 -> 134
(1-loop)
100 ->
101 -> 103 -> 107 -> 115 -> 122 -> 119 -> 130 -> 134
(1-loop)
...
Well,
is there something interesting in this exhaustive search or will those chains
always end in loops and fixed points? Could all fixed points be reached? The
chain 19-134 has 19 terms: could a chain be as long as one wishes? (*)
The
first such fixed points are listed here:
112, 121, 134, 143, 156, 165, 178, 187, 211, 314, 336, 341, 358, 363,
385, 413, 431, 516, 538, 561, 583, 615, 633, 651, 718, 781, 817, 835, 853, 871,
1012...
The sequence of integers
looping on them-selves with the rule “Add odd digits, subtract even ones” will
be submitted soon to Neil Sloane’s OEIS; it starts like this (and includes all
terms of A036301, of course):
0,11,13,17,18,25,28,54,55,64,65,112,121,134,137,143,148,155,156,165,166,173...
Those are the numbers
highlighted in black, in the first
column of the above table (and in blue on r.e.s.’s diagram at very bottom
of page).
Best,
É.
(the
other way rule -add even digits, subtract odd ones- is there)
_____________________________
(*) Franklin T. Adams-Watters:
(...)
Every chain of this sort will end in a loop of some
sort. Consider a number N starting with n+2 “8” digits, followed by n 0's, with
n >= 1. Any number less than this which transforms to something larger will still
transform to number starting with n+2 8's, and then the next term(s) will be
smaller until a number less than N is reached.
The chain thus cannot grow to infinity, so it must
eventually loop.
Certainly the length of a chain increases without
limit: start with a large enough sequence of digits with the same parity (not
0's or 9's), and you will get a large number of steps in the same direction.
The interesting question is whether there are arbitrarily large loops. My guess
is that there are not.
Essentially the same argument applies to adding even
digits and subtracting odd; just look at 9's instead of 8's.
(...)
_____________________________
Joshua Zucker:
I played around with this a little, and found: up to
1000000, the longest cycle is 11, and cycles of that length include the
following numbers (I didn't check to be sure that I have only one
representative of each, though it looks like there's enough distance between
them that I'm OK):
18201 81201 108201 126201 144201 162201 180201 216201
238201 261201 283201 328201 346201 364201 382201 414201 436201 441201 458201
463201 485201 548201 566201 584201 612201 621201 634201 643201 656201 665201
678201 687201 768201 786201 801201 810201 823201 832201 845201 854201 867201
876201 889201 898201 988201
Why do they all end in 201???
Here is one element of each cycle of length 10:
128201 146201 164201 182201 218201 281201 348201
366201 384201 416201 438201 461201 483201 568201 586201 614201 636201 641201
658201 663201 685201 687980 788201 812201 821201 834201 843201 856201 865201
867980 878201 887201
An example of one full cycle, length 11:
18191 -> 18195 -> 18203 -> 18197 -> 18207 -> 18205 -> 18201
-> 18193 -> 18199 -> 18211 -> 18204 -> 18191
and length 10:
128191 -> 128193 -> 128197 -> 128205 -> 128199 -> 128209 ->
128207 -> 128203 -> 128195 -> 128201 -> 128191
A cycle of length 9 that doesn't involve ending in 201
(uses 101 instead):
6095 -> 6103 -> 6101 -> 6097 -> 6107 -> 6109 ->
6113 -> 6112 -> 6106 -> 6095
and one that does:
16193 -> 16201 -> 16195 -> 16205 -> 16203 -> 16199
-> 16213 -> 16210 -> 16204 -> 16193
and here's a few really unusual ones:
600012 600005 600004 599994 600031 600029 600030
600027 600026 380037 380042 380031 380030 380028 380013 380012 380006 379995
687974 687979 687997 688015 687999 688019 688007 687992 688001 687980 867974
867979 867997 868015 867999 868019 868007 867992 868001 867980
(all the rest of length 9 or
more under a million contain a number ending in 101 or 201).
Of course, with more digits there will be trivial
variations of these weird ones, e.g. which replace a leading 3 with 111, or a
leading 6 with 42 or 24 or 222.
_____________________________
On december 4th,
2006, I’ve asked on rec.puzzles:
could someone be so nice to compute
for me two hundred more terms of
this sequence:
0,11,13,17,18,25,28,54,55,64,65...
Explanations can be found
there:
http://www.cetteadressecomportecinquantesignes.com/134.htm
I'll mention yr name in Neil
Sloane's OEIS if the sequence is accepted.
Thanks,
Éric A.
[crossposted to alt.math.recreational
an hour ago]
Got this answer a few minutes later, from “Barry” and “Jongware”:
0 11 13 17 18 25 28 54 55 64 65 112 121
134 137 143 148 155 156 165 166 173
178 184 187 198 200 209 211 216 231 233
234 237 244 245 270 275 280 285 314
336 341 358 363 385 396 402 407 410 413
429 431 432 450 451 453 457 460 465
516 538 561 583 594 604 615 633 651 671
673 676 677 685 718 781 792 806 817
835 853 871 1012 1021 1034 1037 1043
1048 1055 1056 1065 1066 1073 1078 1084
1087 1102 1120 1135 1138 1145 1148 1201
1210 1217 1223 1224 1232 1235 1242
1245 1253 1254 1260 1267 1276 1289 1298
1304 1322 1340 1378 1381 1397 1403
1408 1415 1417 1418 1422 1425 1430 1447
1452 1469 1474 1496 1506 1524 1542
1560 1578 1583 1584 1595 1605 1606 1615
1616 1627 1649 1650 1672 1694 1708
1726 1744 1762 1780 1793 1804 1807 1810
1813 1829 1870 1892 1928 1946 1964
1982 2009 2011 2016 2031 2033 2034 2037
2044 2045 2070 2075 2080 2085 2101
2110 2117 2123 2124 2132 2135 2142 2145
2153 2154 2160 2167 2176 2189 2198
2207 2210 2213 2229 2231 2232 2250 2251
2253 2257 2260 2265 2312 2321 2334
2337 2343 2348 2355 2356 2365 2366 2373 2378
2384 2387 2394 2400 2415 2433
2451 2471 2473 2476 2477 2485 2514 2536
2541 2558 2563 2585 2592 2602 2617
2635 2653 2671 2716 2738 2761 2783 2790
2804 2819 2837 2855 2873 2891 2918
2981 3014 3036 3041 3058 3063 3085 3104
3122 3140 3178 3181 3212 3221 3234
3237 3243 3248 3255 3256 3265 3266 3273
3278 3284 3287 3306 3324 3342 3360
3378 3383 3384 3401 3410 3417 3423 3424
3432 3435 3442 3445 3453 3454 3460
3467 3476 3489 3498 3508 3526 3544 3562
3580 3597 3603 3608 3615 3617 3618
3621 3622 3625 3630 3647 3652 3669 3674
3696 3728 3746 3764 3782 3795 3805
3806 3812 3813 3819 3824 3827 3849 3850
3872 3894 3948 3966 3984 4007 4010
4013 4029 4031 4032 4050 4051 4053 4057
4060 4065 4097 4103 4108 4109 4115
4118 4125 4130 4147 4152 4169 4174 4196
4215 4233 4251 4271 4273 4276 4277
4285 4301 4310 4317 4323 4324 4332 4335
4342 4345 4353 4354 4360 4367 4376
4389 4398 4417 4435 4453 4471 4512 4521
4534 4537 4543 4548 4555 4556 4565
4566 4573 4578 4584 4587 4590 4600 4619
4637 4655 4673 4691 4714 4736 4741
4758 4763 4775 4785 4788 4790 4802 4839
4857 4875 4893 4916 4938 4961 4983
5016 5038 5061 5083 5106 5124 5142 5160
5178 5183 5184 5214 5236 5241 5258
5263 5285 5308 5326 5344 5362 5380 5412
5421 5434 5437 5443 5448 5455 5456
5465 5466 5473 5478 5484 5487 5528 5546
5564 5582 5601 5610 5617 5623 5624
5632 5635 5642 5645 5653 5654 5660 5667
5676 5689 5698 5748 5766 5784 5803
5825 5830 5847 5852 5869 5874 5896 5968
5986 5994 6007 6008 6011 6013 6015
6033 6051 6071 6073 6076 6077 6085 6095
6097 6101 6103 6105 6106 6107 6109
6112 6113 6127 6149 6150 6172 6194 6217
6235 6253 6271 6297 6303 6305 6307
6308 6310 6311 6325 6330 6347 6352 6369
6374 6396 6419 6437 6455 6473 6491
6501 6510 6517 6523 6524 6532 6535 6542
6545 6553 6554 6560 6567 6576 6589
6598 6599 6616 6639 6657 6675 6693 6712
6721 6734 6737 6743 6748 6755 6756
6765 6766 6773 6778 6784 6787 6797 6814
6859 6877 6895 6914 6936 6941 6958
6963 6985 7018 7081 7108 7126 7144 7162
7180 7216 7238 7261 7283 7328 7346
7364 7382 7414 7436 7441 7458 7463 7485
7548 7566 7584 7612 7621 7634 7637
7643 7648 7655 7656 7665 7666 7673 7678
7684 7687 7768 7786 7801 7810 7817
7823 7824 7832 7835 7842 7845 7853 7854
7860 7867 7876 7889 7898 7988 7998
8003 8009 8010 8013 8015 8017 8035 8053
8071 8093 8095 8097 8099 8101 8103
8105 8106 8107 8109 8111 8129 8170 8192
8219 8237 8255 8273 8291 8295 8297
8299 8301 8303 8305 8306 8307 8309 8312
8313 8327 8349 8350 8372 8394 8399
8412 8439 8457 8475 8493 8497 8499 8501
8503 8505 8507 8508 8510 8511 8525
8530 8547 8552 8569 8574 8596 8597 8610
8659 8677 8695 8701 8710 8717 8723
8724 8732 8735 8742 8745 8753 8754 8760
8767 8776 8789 8798 8879 8897 8912
8921 8934 8937 8943 8948 8955 8956 8965
8966 8973 8978 8984 8987 9128 9146
9164 9182 9218 9281 9348 9366 9384 9416
9438 9461 9483 9568 9586 9614 9636
9641 9658 9663 9685 9788 9812 9821 9834
9837 9843 9848 9855 9856 9865 9866
9873 9878 9884 9887 10012 10021 10034
10037 10043 10048 10055 10056 10065
10066 10073 10078 10084 10087 10102 10120
10135 10138 10145 10148 10201
10210 10217 10223 10224 10232 10235
10242 10245 10253 10254 10260 10267
10276 10289 10298 10304 10322 10340
10378 10381 10397 10403 10408 10415
10417 10418 10422 10425 10430 10447
10452 10469 10474 10496 10506 10524
10542 10560 10578 10583 10584 10595
10605 10606 10615 10616 10627 10649
10650 10672 10694 10708 10726 10744
10762 10780 10793 10804 10807 10810
10813 10829 10870 10892 10928 10946
10964 10982 11002 11020 11035 11038
11045 11048 11114 11136 11141 11158 11163
11185 11200 11211 11213 11217
11218 11225 11228 11254 11255 11264
11265 11316 11338 11361 11383 11398
11404 11409 11411 11416 11431 11433
11434 11437 11444 11445 11470 11475
11480 11485 11518 11581 11596 11606
11607 11610 11613 11629 11631 11632
11650 11651 11653 11657 11660 11665
11794 11808 11815 11833 11851 11871
11873 11876 11877 11885 12001 12010
12017 12023 12024 12032 12035 12042
12045 12053 12054 12060 12067 12076
12089 12098 12100 12111 12113 12117
12118 12125 12128 12154 12155 12164 12165
12197 12203 12208 12213 12214
12225 12230 12247 12252 12269 12274
12296 12302 12320 12335 12338 12345
12348 12395 12405 12408 12411 12427
12449 12450 12472 12494 12504 12522
12540 12578 12581 12593 12606 12607
12609 12611 12629 12670 12692 12706
12724 12742 12760 12778 12783 12784
12791 12793 12801 12804 12805 12807
12809 12811 12890 12908 12926 12944
12962 12980 13004 13022 13040 13078
13081 13116 13138 13161 13183 13202
13220 13235 13238 13245 13248 13318
13381 13400 13411 13413 13417 13418 13425
13428 13454 13455 13464 13465
13598 13608 13609 13611 13616 13631
13633 13634 13637 13644 13645 13670
13675 13680 13685 13807 13810 13813
13829 13831 13832 13850 13851 13853
13857 13860 13865 13997 14003 14008
14015 14018 14025 14026 14030 14047
14052 14069 14074 14096 14109 14111
14116 14131 14133 14134 14137 14144
14145 14170 14175 14180 14185 14195
14205 14206 14207 14209 14212 14213
14227 14249 14250 14272 14294 14300
14311 14313 14317 14318 14325 14328
14354 14355 14364 14365 14393 14399 14403
14404 14405 14407 14417 14418
14429 14470 14492 14502 14520 14535
14538 14545 14548 14591 14593 14597
14601 14602 14603 14605 14607 14609
14612 14615 14690 14704 14722 14740
14778 14781 14789 14794 14795 14803
14806 14813 14906 14924 14942 14960
14978 14983 14984 15006 15024 15042
15060 15078 15083 15084 15118 15181
15204 15222 15240 15278 15281 15402
15420 15435 15438 15445 15448 15600
15611 15613 15617 15618 15625 15628
15654 15655 15664 15665 15809 15811
15816 15831 15833 15834 15837 15844 15845
15870 15875 15880 15885 15995
16005 16006 16012 16013 16024 16027
16049 16050 16072 16094 16107 16110
16113 16129 16131 16132 16150 16151
16153 16157 16160 16165 16193 16195
16199 16201 16203 16204 16205 16207
16210 16213 16229 16270 16292 16309
16311 16316 16331 16333 16334 16337
16344 16345 16370 16375 16380 16385
16391 16399 16402 16409 16412 16415
16490 16500 16511 16513 16517 16518
16525 16528 16554 16555 16564 16565
16589 16590 16591 16595 16599 16600
16601 16603 16607 16609 16614 16617 16702
16720 16735 16738 16745 16748
16774 16779 16787 16788 16789 16792
16797 16801 16808 16815 16904 16922
16940 16978 16981 17008 17026 17044
17062 17080 17206 17224 17242 17260
17278 17283 17284 17404 17422 17440
17478 17481 17602 17620 17635 17638
17645 17648 17800 17811 17813 17817
17818 17825 17828 17854 17855 17864
17865 17993 18006 18007 18011 18022
18029 18070 18092 18115 18133 18151
18171 18173 18176 18177 18185 18191
18193 18195 18197 18199 18201 18203
18204 18205 18207 18209 18211 18290 18307
18310 18313 18329 18331 18332
18350 18351 18353 18357 18360 18365
18509 18511 18516 18531 18533 18534
18537 18544 18545 18570 18575 18580
18585 18593 18603 18700 18711 18713
18717 18718 18725 18728 18754 18755
18764 18765 18772 18777 18786 18787
18791 18796 18799 18801 18810 18817
18902 18920 18935 18938 18945 18948
19028 19046 19064 19082 19208 19226
19244 19262 19280 19406 19424 19442
19460 19478 19483 19484 19604 19622
19640 19678 19681 19802 19820 19835
19838 19845 19848 20009 20011 20016 20031
20033 20034 20037 20044 20045
20070 20075 20080 20085 20101 20110
20117 20123 20124 20132 20135 20142
20145 20153 20154 20160 20167 20176
20189 20198 20207 20210 20213 20229
20231 20232 20250 20251 20253 20257
20260 20265 20312 20321 20334 20337
20343 20348 20355 20356 20365 20366
20373 20378 20384 20387 20394 20400
20415 20433 20451 20471 20473 20476
20477 20485 20514 20536 20541 20558
20563 20585 20592 20602 20617 20635
20653 20671 20716 20738 20761 20783
20790 20804 20819 20837 20855 20873 20891
20918 20981 21001 21010 21017
21023 21024 21032 21035 21042 21045
21053 21054 21060 21067 21076 21089
21098 21100 21111 21113 21117 21118
21125 21128 21154 21155 21164 21165
21197 21203 21208 21213 21214 21225
21230 21247 21252 21269 21274 21296
21302 21320 21335 21338 21345 21348
21395 21405 21408 21411 21427 21449
21450 21472 21494 21504 21522 21540
21578 21581 21593 21606 21607 21609
21611 21629 21670 21692 21706 21724
21742 21760 21778 21783 21784 21791
21793 21801 21804 21805 21807 21809 21811
21890 21908 21926 21944 21962
21980 22007 22010 22013 22029 22031
22032 22050 22051 22053 22057 22060
22065 22097 22103 22108 22109 22115
22118 22125 22130 22147 22152 22169
22174 22196 22215 22233 22251 22271
22273 22276 22277 22285 22301 22310
22317 22323 22324 22332 22335 22342
22345 22353 22354 22360 22367 22376
22389 22398 22417 22435 22453 22471
22512 22521 22534 22537 22543 22548
22555 22556 22565 22566 22573 22578
22584 22587 22590 22600 22619 22637
22655 22673 22691 22714 22736 22741 22758
22763 22775 22785 22788 22790
22802 22839 22857 22875 22893 22916
22938 22961 22983 23012 23021 23034
23037 23043 23048 23055 23056 23065
23066 23073 23078 23084 23087 23102
23120 23135 23138 23145 23148 23201
23210 23217 23223 23224 23232 23235
23242 23245 23253 23254 23260 23267
23276 23289 23298 23304 23322 23340
23378 23381 23397 23403 23408 23415
23417 23418 23422 23425 23430 23447
23452 23469 23474 23496 23506 23524
23542 23560 23578 23583 23584 23595
23605 23606 23615 23616 23627 23649 23650
23672 23694 23708 23726 23744
23762 23780 23793 23804 23807 23810
23813 23829 23870 23892 23928 23946
23964 23982 23994 24004 24005 24009
24012 24015 24033 24051 24071 24073
24076 24077 24085 24095 24097 24101
24103 24105 24106 24107 24109 24112
24113 24127 24149 24150 24172 24194
24217 24235 24253 24271 24297 24303
24305 24307 24308 24310 24311 24325
24330 24347 24352 24369 24374 24396
24419 24437 24455 24473 24491 24501
24510 24517 24523 24524 24532 24535
24542 24545 24553 24554 24560 24567 24576
24589 24598 24599 24616 24639
24657 24675 24693 24712 24721 24734
24737 24743 24748 24755 24756 24765
24766 24773 24778 24784 24787 24797
24814 24859 24877 24895 24914 24936
24941 24958 24963 24985 25014 25036
25041 25058 25063 25085 25104 25122
25140 25178 25181 25212 25221 25234
25237 25243 25248 25255 25256 25265
25266 25273 25278 25284 25287 25306
25324 25342 25360 25378 25383 25384
25401 25410 25417 25423 25424 25432
25435 25442 25445 25453 25454 25460
25467 25476 25489 25498 25508 25526 25544
25562 25580 25597 25603 25608
25615 25617 25618 25621 25622 25625
25630 25647 25652 25669 25674 25696
25728 25746 25764 25782 25795 25805
25806 25812 25813 25819 25824 25827
25849 25850 25872 25894 25948 25966
25984 25992 26002 26005 26011 26017
26035 26053 26071 26093 26095 26097
26099 26101 26103 26105 26106 26107
26109 26111 26129 26170 26192 26219
26237 26255 26273 26291 26295 26297
26299 26301 26303 26305 26306 26307
26309 26312 26313 26327 26349 26350
26372 26394 26399 26412 26439 26457 26475
26493 26497 26499 26501 26503
26505 26507 26508 26510 26511 26525
26530 26547 26552 26569 26574 26596
26597 26610 26659 26677 26695 26701
26710 26717 26723 26724 26732 26735
26742 26745 26753 26754 26760 26767
26776 26789 26798 26879 26897 26912
26921 26934 26937 26943 26948 26955
26956 26965 26966 26973 26978 26984
26987 27016 27038 27061 27083 27106
27124 27142 27160 27178 27183 27184
27214 27236 27241 27258 27263 27285
27308 27326 27344 27362 27380 27412
27421 27434 27437 27443 27448 27455 27456
27465 27466 27473 27478 27484
27487 27528 27546 27564 27582 27601
27610 27617 27623 27624 27632 27635
27642 27645 27653 27654 27660 27667
27676 27689 27698 27748 27766 27784
27803 27825 27830 27847 27852 27869
27874 27896 27968 27986 27990 28004
28007 28013 28019 28037 28055 28073
28091 28093 28095 28097 28099 28101
28103 28105 28107 28109 28190 28239
28257 28275 28293 28295 28297 28299
28301 28303 28305 28307 28329 28370
28392 28459 28477 28495 28497 28499
28501 28503 28505 28527 28549 28550 28572
28594 28679 28697 28699 28701
28703 28725 28730 28747 28752 28769
28774 28796 28899 28901 28910 28917
28923 28924 28932 28935 28942 28945
28953 28954 28960 28967 28976 28989
28998 29018 29081 29108 29126 29144
29162 29180 29216 29238 29261 29283
29328 29346 29364 29382 29414 29436
29441 29458 29463 29485 29548 29566
29584 29612 29621 29634 29637 29643
29648 29655 29656 29665 29666 29673
29678 29684 29687 29768 29786 29801
29810 29817 29823 29824 29832 29835
29842 29845 29853 29854 29860 29867 29876
29889 29898 29988
... which are
the 2,335 used values in the first 30,001 integers (0 to 30,000). Want more?
_____________________________
No, thanks 30001 times, Barry and Jongware!
É.
_____________________________
On december 5th I received
those beautiful diagrams from r.e.s. In blue are the “self-loopers”:
r.e.s. asks on rec.puzzles:
Some questions that come to mind for
this discrete dynamical system (just throwing these out -- haven't really
thought much about them):
(1) Are there infinitely-many
attractors?
(2) If there are infinitely-many
attractors, which ones (if any, besides 0) attract only finitely-many points?
All of them?
(3) What is the set of attractor
periods? (E.g., we have 1,4,6,... -- but
are there none of period 2,3,5?)
(4) In listing the attractor-points in
increasing order as you've done, will some of the attractors overlap? (E.g., is
there any occurrence of ...,a,b,c,...
where a,c is part of one attractor and b is part of
another?)
(5) How complicated does the structure
of the attractor basins become? (E.g., if there exists an attractor that
attracts infinitely-many points, do its "tributaries" exhibit
infinitely-many "forks"?)
--r.e.s.
Any takers?
Send yr comments here.
________________________
Received December 6th, from Jongware (a.k.a. Theunis
de Jong):
Just a thought on the longest chain :)
Rewrote my proggie
to return the length of the chain per number and show the longest one. In the
range up to 1,000,000, the longest is a staggering 289 numbers long:
883867 -> 883847 -> 883829 ->
883815 -> 883800 -> 883779 -> 883789 -> 883784 -> 883766 ->
883748 -> 883730 -> 883727 -> 883726 -> 883712 -> 883705 ->
883704 -> 883694 -> 883680 -> 883653 -> 883642 -> 883617 ->
883606 -> 883581 -> 883566 -> 883546 -> 883528 -> 883510 ->
883503 -> 883498 -> 883482 -> 883455 -> 883448 -> 883419 ->
883412 -> 883394 -> 883389 -> 883380 -> 883362 -> 883344 ->
883326 -> 883308 -> 883290 -> 883284 -> 883257 -> 883254 ->
883240 -> 883221 -> 883205 -> 883195 -> 883197 -> 883201 ->
883187 -> 883174 -> 883165 -> 883152 -> 883143 -> 883130 ->
883121 -> 883108 -> 883088 -> 883059 -> 883060 -> 883041 ->
883025 -> 883015 -> 883008 -> 882987 -> 882977 -> 882982 ->
882963 -> 882951 -> 882948 -> 882927 -> 882923 -> 882915 ->
882912 -> 882902 -> 882891 -> 882875 -> 882861 -> 882830 ->
882807 -> 882788 -> 882761 -> 882745 -> 882735 -> 882732 ->
882722 -> 882707 -> 882703 -> 882695 -> 882685 -> 882658 ->
882631 -> 882611 -> 882589 -> 882577 -> 882578 -> 882564 ->
882541 -> 882525 -> 882515 -> 882508 -> 882487 -> 882464 ->
882432 -> 882411 -> 882391 -> 882386 -> 882357 -> 882354 ->
882340 -> 882321 -> 882305 -> 882295 -> 882289 -> 882270 ->
882257 -> 882249 -> 882234 -> 882213 -> 882197 -> 882196 ->
882182 -> 882155 -> 882148 -> 882119 -> 882112 -> 882094 ->
882081 -> 882056 -> 882037 -> 882029 -> 882018 -> 881993 ->
881999 -> 882011 -> 881995 -> 882003 -> 881988 -> 881966 -> 881948
-> 881930 -> 881927 -> 881926 -> 881912 -> 881905 -> 881904
-> 881894 -> 881876 -> 881854 -> 881832 -> 881810 -> 881788
-> 881764 -> 881746 -> 881728 -> 881710 -> 881703 -> 881698
-> 881678 -> 881656 -> 881634 -> 881612 -> 881590 -> 881589
-> 881580 -> 881562 -> 881544 -> 881526 -> 881508 -> 881490
-> 881480 -> 881453 -> 881442 -> 881417 -> 881406 -> 881381
-> 881362 -> 881342 -> 881324 -> 881306 -> 881288 -> 881255
-> 881248 -> 881219 -> 881212 -> 881194 -> 881185 -> 881168
-> 881140 -> 881122 -> 881104 -> 881086 -> 881057 -> 881054
-> 881040 -> 881021 -> 881005 -> 880995 -> 881002 -> 880985
-> 880975 -> 880980 -> 880965 -> 880957 -> 880962 -> 880947
-> 880943 -> 880935 -> 880936 -> 880926 -> 880911 -> 880906
-> 880893 -> 880881 -> 880850 -> 880831 -> 880811 -> 880789
-> 880781 -> 880765 -> 880755 -> 880756 -> 880746 -> 880727
-> 880723 -> 880715 -> 880712 -> 880702 -> 880691 -> 880679
-> 880673 -> 880661 -> 880634 -> 880611 -> 880591 -> 880590
-> 880588 -> 880561 -> 880545 -> 880535 -> 880532 -> 880522 ->
880507 -> 880503 -> 880495 -> 880489 -> 880470 -> 880457 ->
880449 -> 880434 -> 880413 -> 880397 -> 880400 -> 880380 ->
880359 -> 880360 -> 880341 -> 880325 -> 880315 -> 880308 ->
880287 -> 880268 -> 880236 -> 880215 -> 880203 -> 880188 ->
880157 -> 880154 -> 880140 -> 880121 -> 880105 -> 880095 ->
880093 -> 880089 -> 880074 -> 880061 -> 880040 -> 880020 -> 880002 -> 879984 -> 879989 ->
880007 -> 879998 -> 880016 -> 879995 -> 880026 -> 880002 (9-loop,
289 iterations)
The max length is logically somehow linked
to the number of decimals in the starting number (but *how* is entirely beyond
my comprehension :)
________________________
Beautiful! Thanks again, Theunis, and to all other contributors as well!
This sequence is now in the OEIS as A124176.