...
Had we chosen the other way rule (add
even digits, subtract odd ones), would we have this table:
0
-> 0 (a 1-loop or fixed point)
1 -> 0 (1-loop)
2 -> 4 -> 8 -> 16 -> 21 -> 22 -> 26 -> 34 -> 35 -> 27
-> 22 (a 5-loop
and a 10-chain)
3 -> 0 (1-loop)
4 -> 8 -> 16 -> 21 -> 22 -> 26 -> 34 -> 35 -> 27
-> 22
(5-loop)
5 -> 0 (1-loop)
6 -> 12 -> 13 -> 9 -> 0 (1-loop)
7 -> 0 (1-loop)
8 -> 16 -> 21 -> 22 -> 26 -> 34 -> 35 -> 27 -> 22 (5-loop)
9 -> 0 (1-loop)
10
-> 9 -> 0 (1-loop)
11
-> 9 -> 0 (1-loop)
12
-> 13 -> 9 -> 0 (1-loop)
13
-> 9 -> 0 (1-loop)
14
-> 17 -> 9 -> 0 (1-loop)
15
-> 9 -> 0 (1-loop)
16
-> 21 -> 22 -> 26 -> 34
-> 35 -> 27 -> 22 (5-loop)
17
-> 9 -> 0 (1-loop)
18
-> 25 -> 22 -> 26 -> 34
-> 35 -> 27 -> 22 (5-loop)
19
-> 9 -> 0 (1-loop)
20
-> 22 -> 26 -> 34 -> 35
-> 27 -> 22 (5-loop)
21
-> 22 -> 26 -> 34 -> 35
-> 27 -> 22 (5-loop)
22 -> 26 -> 34 -> 35
-> 27 -> 22 (5-loop)
23
-> 22 -> 26 -> 34 -> 35
-> 27 -> 22 (5-loop)
24
-> 30 -> 27 -> 22 -> 26
-> 34 -> 35 -> 27 (5-loop)
25
-> 22 -> 26 -> 34 -> 35
-> 27 -> 22 (5-loop)
26 -> 34 -> 35 -> 27
-> 22 -> 26 (5-loop)
27 -> 22 -> 26 -> 34
-> 35 -> 27 (5-loop)
28
-> 38 -> 43 -> 44 -> 52
-> 49 -> 44 (3-loop)
29
-> 22 -> 26 -> 34 -> 35
-> 27 -> 22 (5-loop)
30
-> 27 -> 22 -> 26 -> 34
-> 35 -> 27 (5-loop)
31
-> 27 -> 22 -> 26 -> 34
-> 35 -> 27 (5-loop)
32
-> 31 -> 27 -> 22 -> 26
-> 34 -> 35 -> 27 (5-loop)
33
-> 27 -> 22 -> 26 -> 34
-> 35 -> 27 (5-loop)
34 -> 35 -> 27 -> 22
-> 26 -> 34 (5-loop)
35 -> 27 -> 22 -> 26
-> 34 -> 35 (5-loop)
36
-> 39 -> 27 -> 22 -> 26
-> 34 -> 35 -> 27 (5-loop)
37
-> 27 -> 22 -> 26 -> 34
-> 35 -> 27 (5-loop)
38
-> 43 -> 44 -> 52 -> 49
-> 44 (3-loop)
39
-> 27 -> 22 -> 26 -> 34
-> 35 -> 27 (5-loop)
40
-> 44 -> 52 -> 49 -> 44 (3-loop)
41
-> 44 -> 52 -> 49 -> 44 (3-loop)
42
-> 48 -> 60 -> 66 -> 78
-> 79 -> 63 -> 66 (4-loop)
43
-> 44 -> 52 -> 49 -> 44 (3-loop)
44 -> 52 -> 49 -> 44 (3-loop)
45
-> 44 -> 52 -> 49 -> 44 (3-loop)
46
-> 56 -> 57 -> 45 -> 44
-> 52 -> 49 -> 44 (3-loop)
47
-> 44 -> 52 -> 49 -> 44 (3-loop)
48
-> 60 -> 66 -> 78 -> 79
-> 63 -> 66 (4-loop)
49 -> 44 -> 52 -> 49 (3-loop)
50
-> 45 -> 44 -> 52 -> 49
-> 44 (3-loop)
51
-> 47 -> 44 -> 52 -> 49
-> 44 (3-loop)
52 -> 49 -> 44 -> 52 (3-loop)
53
-> 51 -> 47 -> 44 -> 52
-> 49 -> 44 (3-loop)
54
-> 53 -> 51 -> 47 -> 44
-> 52 -> 49 -> 44 (3-loop)
55
-> 45 -> 44 -> 52 -> 49
-> 44 (3-loop)
56
-> 57 -> 45 -> 44 -> 52
-> 49 -> 44 (3-loop)
57
-> 45 -> 44 -> 52 -> 49
-> 44 (3-loop)
58
-> 61 -> 66 -> 78 -> 79
-> 63 -> 66 (4-loop)
59
-> 45 -> 44 -> 52 -> 49
-> 44 (3-loop)
60
-> 66 -> 78 -> 79 -> 63
-> 66 (4-loop)
61
-> 66 -> 78 -> 79 -> 63
-> 66 (4-loop)
62
-> 70 -> 63 -> 66 -> 78
-> 79 -> 63 (4-loop)
63 -> 66 -> 78 -> 79
-> 63 (4-loop)
64
-> 74 -> 71 -> 63 -> 66
-> 78 -> 79 -> 63 (4-loop)
65
-> 66 -> 78 -> 79 -> 63
-> 66 (4-loop)
66 -> 78 -> 79 -> 63
-> 66 (4-loop)
67
-> 66 -> 78 -> 79 -> 63
-> 66 (4-loop)
68
-> 82 -> 92 -> 85 -> 88
-> 104 -> 107 -> 99 -> 81 -> 88 (5-loop)
69
-> 56 -> 57 -> 45 -> 44
-> 52 -> 49 -> 44 (3-loop)
70
-> 63 -> 66 -> 78 -> 79
-> 63 (4-loop)
71
-> 65 -> 66 -> 78 -> 79
-> 63 -> 66 (4-loop)
72
-> 67 -> 66 -> 78 -> 79
-> 63 -> 66 (4-loop)
73
-> 63 -> 66 -> 78 -> 79
-> 63 (4-loop)
74
-> 71 -> 65 -> 66 -> 78
-> 79 -> 63 -> 66 (4-loop)
75
-> 63 -> 66 -> 78 -> 79
-> 63 (4-loop)
76
-> 75 -> 63 -> 66 -> 78
-> 79 -> 63 (4-loop)
77
-> 63 -> 66 -> 78 -> 79
-> 63 (4-loop)
78 -> 79 -> 63 -> 66
-> 78 (4-loop)
79 -> 63 -> 66 -> 78
-> 79 (4-loop)
80
-> 88 -> 104 -> 107 ->
99 -> 81 -> 88 (5-loop)
81 -> 88 -> 104 ->
107 -> 99 -> 81 (5-loop)
82
-> 92 -> 85 -> 88 -> 104
-> 107 -> 99 -> 81 -> 88 (5-loop)
83
-> 88 -> 104 -> 107 ->
99 -> 81 -> 88 (5-loop)
84
-> 96 -> 93 -> 81 -> 88
-> 104 -> 107 -> 99 -> 81 (5-loop)
85
-> 88 -> 104 -> 107 ->
99 -> 81 -> 88 (5-loop)
86 ->
100 -> 99 -> 81 -> 88 ->
104 -> 107 -> 99 (5-loop)
87
-> 88 -> 104 -> 107 ->
99 -> 81 -> 88 (5-loop)
88 -> 104 -> 107 ->
99 -> 81 -> 88 (5-loop)
89
-> 88 -> 104 -> 107 ->
99 -> 81 -> 88 (5-loop)
90
-> 83 -> 88 -> 104 ->
107 -> 99 -> 81 -> 88 (5-loop)
91
-> 81 -> 88 -> 104 ->
107 -> 99 -> 81 (5-loop)
92
-> 85 -> 88 -> 104 ->
107 -> 99 -> 81 -> 88 (5-loop)
93
-> 81 -> 88 -> 104 -> 107
-> 99 -> 81 (5-loop)
94
-> 89 -> 88 -> 104 ->
107 -> 99 -> 81 -> 88 (5-loop)
95
-> 81 -> 88 -> 104 ->
107 -> 99 -> 81 (5-loop)
96
-> 93 -> 81 -> 88 -> 104
-> 107 -> 99 -> 81 (5-loop)
97
-> 81 -> 88 -> 104 ->
107 -> 99 -> 81 (5-loop)
98 -> 97 -> 81 -> 88 -> 104 -> 107 -> 99 -> 81 (5-loop)
99 -> 81 -> 88 -> 104
-> 107 -> 99 (5-loop)
100 -> 99
-> 81 -> 88 -> 104 -> 107 -> 99 (5-loop)
...
etc.
Same questions arise:
Is
there something interesting in this too? Will all such chains always end in
loops or fixed points (a zero-loop is
a fixed point)? Could all fixed points be reached, with this rule? The chain
starting with “2” loops after 10 terms: could a chain be as long as one wishes?
First
1-loop numbers (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 even digits, subtract odd ones” will
be submitted soon to Neil Sloane’s OEIS; it starts like this (and includes all
terms of A036301, of course):
0,22,26,27,34,35,44,49,52,63,66,78,79,81,88,99,104,107,108,112,115,121,126...
Those are the numbers
highlighted in black, in the first column
of the above table.
Best,
É.
(the
other way rule -add odd digits, subtract
even ones- is there)
____________________________
[Jongware] :
> Want more?
[Me, Eric] :
... Geee Jongware,
you and Barry are too much ! 30000
thanks !
I'll update my page in a couple of minutes...
(Am I too demanding if I ask for the same job here:
http://www.cetteadressecomportecinquantesignes.com/22.htm)
[Jongware] :
>
S'thing like this?
0
22 26 27 34 35 44 49 52 63 66 78 79 81 88 99 104 107 108 112 115 121 126
133
134 143 144 151 156 165 178 187 211 224 229 232 233 283 290 314 336 341
358
363 385 413 431 467 470 489 492 516 538 561 583 615 633 651 669 672 673
687
694 695 718 781 817 835 853 871 889 891 896 901 1008 1012 1015 1021 1026
1033
1034 1043 1044 1051 1056 1065 1078 1087 1102 1120 1142 1143 1146 1151
1154
1164 1165 1172 1201 1210 1223 1228 1232 1239 1245 1246 1254 1257 1264
1267
1275 1276 1282 1289 1293 1298 1304 1318 1321 1322 1340 1362 1366 1367
1374
1381 1384 1392 1403 1425 1430 1447 1452 1469 1474 1484 1488 1492 1494
1496
1499 1506 1507 1524 1542 1560 1582 1583 1586 1594 1605 1627 1649 1650
1672
1686 1690 1692 1694 1696 1698 1702 1705 1708 1726 1744 1762 1780 1807
1829
1870 1888 1892 1898 1899 1904 1911 1928 1946 1964 1982 2011
2024 2029
2032
2033 2083 2090 2101 2110 2123 2128 2132 2139 2145 2146 2154 2157 2164
2167
2175 2176 2182 2189 2193 2198 2213 2231 2267 2270 2289 2292 2308 2312
2315
2321 2326 2333 2334 2343 2344 2351 2356 2365 2378 2387 2415 2433 2451
2469
2472 2473 2487 2494 2495 2514 2536 2541 2558 2563 2585 2617 2635 2653
2671
2689 2695 2696 2701 2716 2738 2761 2783 2819 2837 2855 2873 2891 2918
2981
3014 3036 3041 3058 3063 3085 3104 3118 3121 3122 3140 3162 3166 3167
3174
3181 3184 3192 3208 3212 3215 3221 3226 3233 3234 3243 3244 3251 3256
3265
3278 3287 3288 3294 3303 3306 3324 3342 3360 3382 3383 3386 3394 3401
3410
3423 3428 3432 3439 3445 3446 3454 3457 3464 3467 3475 3476 3482 3486
3489
3492 3493 3498 3501 3508 3526 3544 3562 3580 3603 3625 3630 3647 3652
3669
3674 3684 3696 3699 3728 3746 3764 3782 3805 3827 3849 3850 3872 3888
3894
3909 3948 3966 3984 4013 4031 4067 4070 4089 4092 4103 4125 4130 4147
4152
4169 4174 4184 4196 4199 4215 4233 4251 4269 4272 4273 4287 4294 4295
4301
4310 4323 4328 4332 4339 4345 4346 4354 4357 4364 4367 4375 4376 4382
4389
4393 4398 4417 4435 4453 4471 4489 4496 4499 4501 4508 4512 4515 4521
4526
4533 4534 4543 4544 4551 4556 4565 4578 4587 4619 4637 4655 4673 4691
4714
4736 4741 4758 4763 4785 4839 4857 4875 4893 4916 4938 4961 4983 5016
5038
5061 5083 5106 5124 5142 5160 5182 5183 5186 5194 5214 5236 5241 5258
5263
5285 5308 5326 5344 5362 5380 5408 5412 5415 5421 5426 5433 5434 5443
5444
5451 5456 5465 5478 5487 5528 5546 5564 5582 5601 5610 5623 5628 5632
5639
5645 5646 5654 5657 5664 5667 5675 5676 5682 5688 5689 5693 5698 5705
5748
5766 5784 5803 5825 5830 5847 5852 5869 5874 5884 5886 5896 5899 5903
5968
5986 6015 6033 6051 6069 6072 6073 6087 6094 6095 6105 6127 6149 6150
6172
6194 6217 6235 6253 6271 6303 6325 6330 6347 6352 6369 6374 6384 6396
6399
6419 6437 6455 6473 6491 6501 6510 6523 6528 6532 6539 6545 6546 6554
6557
6564 6567 6575 6576 6582 6589 6593 6598 6639 6657 6675 6693 6708 6712
6715
6721 6726 6733 6734 6743 6744 6751 6756 6765 6778 6787 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 7488 7501 7548
7566
7584 7608 7612 7615 7621 7626 7633 7634 7643 7644 7651 7656 7665 7678
7687
7768 7786 7801 7810 7823 7828 7832 7839 7845 7846 7854 7857 7864 7867
7875
7876 7882 7889 7893 7898 7988 8017 8035 8053 8071 8107 8129 8170 8192
8219
8237 8255 8273 8291 8305 8327 8349 8350 8372 8394 8439 8457 8475 8493
8503
8525 8530 8547 8552 8569 8574 8584 8596 8599 8659 8677 8695 8701 8710
8723
8728 8732 8739 8745 8746 8754 8757 8764 8767 8775 8776 8782 8789 8793
8798
8879 8897 8908 8912 8915 8921 8926 8933 8934 8943 8944 8951 8956 8965
8978
8987 9128 9146 9164 9182 9218 9281 9348 9366 9384 9416 9438 9461 9483
9568
9586 9614 9636 9641 9658 9663 9685 9788 9808 9812 9815 9821 9826 9833
9834
9843 9844 9851 9856 9865 9878 9887 10008 10012 10015 10021 10026 10033
10034
10043 10044 10051 10056 10065 10078 10087 10102 10120 10142 10143
10146
10151 10154 10164 10165 10172 10201 10210 10223 10228 10232 10239
10245
10246 10254 10257 10264 10267 10275 10276 10282 10289 10293 10298
10304
10318 10321 10322 10340 10362 10366 10367 10374 10381 10384 10392
10403
10425 10430 10447 10452 10469 10474 10484 10488 10492 10494 10496
10499
10506 10507 10524 10542 10560 10582 10583 10586 10594 10605 10627
10649
10650 10672 10686 10690 10692 10694 10696 10698 10702 10705 10708
10726
10744 10762 10780 10807 10829 10870 10888 10892 10898 10899 10904
10911
10928 10946 10964 10982 11002 11020 11042 11043 11046 11051 11054
11064
11065 11072 11114 11136 11141 11158 11163 11185 11200 11222 11226
11227
11234 11235 11244 11249 11252 11263 11266 11278 11279 11281 11288
11295
11303 11304 11316 11338 11361 11383 11411 11424 11429 11432 11433
11483
11490 11518 11581 11613 11631 11667 11670 11689 11692 11815 11833
11851
11869 11872 11873 11887 11894 11895 12001 12010 12023 12028 12032
12039
12045 12046 12054 12057 12064 12067 12075 12076 12082 12089 12093
12098
12100 12122 12126 12127 12134 12135 12144 12149 12152 12163 12166
12178
12179 12203 12225 12230 12247 12252 12269 12274 12284 12296 12298
12299
12300 12302 12320 12342 12343 12346 12351 12354 12364 12365 12372
12405
12427 12449 12450 12472 12494 12496 12498 12500 12502 12504 12518
12521
12522 12540 12562 12566 12567 12574 12581 12584 12592 12607 12629
12670
12692 12694 12696 12698 12700 12702 12704 12706 12724 12742 12760
12782
12783 12786 12794 12809 12890 12892 12894 12896 12898 12900 12902
12904
12906 12908 12926 12944 12962 12980 13004 13018 13021 13022 13040
13062
13066 13067 13074 13081 13084 13092 13116 13138 13161 13183 13202
13220
13242 13243 13246 13251 13254 13264 13265 13272 13318 13381 13400
13422
13426 13427 13434 13435 13444 13449 13452 13463 13466 13478 13479
13481
13488 13499 13504 13611 13624 13629 13632 13633 13683 13690 13813
13831
13867 13870 13889 13892 14003 14025 14030 14047 14052 14069 14074
14084
14096 14099 14111 14124 14129 14132 14133 14183 14190 14205 14227
14249
14250 14272 14294 14300 14322 14326 14327 14334 14335 14344 14349
14352
14363 14366 14378 14379 14407 14429 14470 14492 14498 14502 14504
14506
14507 14510 14520 14542 14543 14546 14551 14554 14564 14565 14572
14609
14690 14696 14700 14702 14704 14718 14721 14722 14740 14762 14766
14767
14774 14781 14784 14792 14894 14900 14906 14924 14942 14960 14982
14983
14986 14994 15006 15024 15042 15060 15082 15083 15086 15094 15118
15181
15204 15218 15221 15222 15240 15262 15266 15267 15274 15281 15284
15292
15402 15420 15442 15443 15446 15451 15454 15464 15465 15472 15600
15622
15626 15627 15634 15635 15644 15649 15652 15663 15666 15678 15679
15681
15688 15695 15704 15811 15824 15829 15832 15833 15883 15890 16005
16027
16049 16050 16072 16094 16113 16131 16167 16170 16189 16192 16207
16229
16270 16292 16311 16324 16329 16332 16333 16383 16390 16409 16490
16500
16522 16526 16527 16534 16535 16544 16549 16552 16563 16566 16578
16579
16698 16702 16707 16708 16714 16715 16720 16742 16743 16746 16751
16754
16764 16765 16772 16896 16904 16906 16908 16909 16912 16918 16921
16922
16940 16962 16966 16967 16974 16981 16984 16992 17008 17026 17044
17062
17080 17206 17224 17242 17260 17282 17283 17286 17294 17404 17418
17421
17422 17440 17462 17466 17467 17474 17481 17484 17492 17602 17620
17642
17643 17646 17651 17654 17664 17665 17672 17800 17822 17826 17827
17834
17835 17844 17849 17852 17863 17866 17878 17879 17881 17888 17891
17904
18007 18029 18070 18092 18115 18133 18151 18169 18172 18173 18187
18194
18195 18209 18290 18313 18331 18367 18370 18389 18392 18511 18524
18529
18532 18533 18583 18590 18700 18722 18726 18727 18734 18735 18744
18749
18752 18763 18766 18778 18779 18898 18902 18907 18911 18912 18920
18942
18943 18946 18951 18954 18964 18965 18972 19028 19046 19064 19082
19208
19226 19244 19262 19280 19406 19424 19442 19460 19482 19483 19486
19494
19604 19618 19621 19622 19640 19662 19666 19667 19674 19681 19684
19692
19802 19820 19842 19843 19846 19851 19854 19864 19865 19872 20011
20024
20029 20032 20033 20083 20090 20101 20110 20123 20128 20132 20139
20145
20146 20154 20157 20164 20167 20175 20176 20182 20189 20193 20198
20213
20231 20267 20270 20289 20292 20308 20312 20315 20321 20326 20333
20334
20343 20344 20351 20356 20365 20378 20387 20415 20433 20451 20469
20472
20473 20487 20494 20495 20514 20536 20541 20558 20563 20585 20617
20635
20653 20671 20689 20695 20696 20701 20716 20738 20761 20783 20819
20837
20855 20873 20891 20918 20981 21001 21010 21023 21028 21032 21039
21045
21046 21054 21057 21064 21067 21075 21076 21082 21089 21093 21098
21100
21122 21126 21127 21134 21135 21144 21149 21152 21163 21166 21178
21179
21203 21225 21230 21247 21252 21269 21274 21284 21296 21298 21299
21300
21302 21320 21342 21343 21346 21351 21354 21364 21365 21372 21405
21427
21449 21450 21472 21494 21496 21498 21500 21502 21504 21518 21521
21522
21540 21562 21566 21567 21574 21581 21584 21592 21607 21629 21670
21692
21694 21696 21698 21700 21702 21704 21706 21724 21742 21760 21782
21783
21786 21794 21809 21890 21892 21894 21896 21898 21900 21902 21904
21906
21908 21926 21944 21962 21980 22013 22031 22067 22070 22089 22092
22103
22125 22130 22147 22152 22169 22174 22184 22196 22199 22215 22233
22251
22269 22272 22273 22287 22294 22295 22301 22310 22323 22328 22332
22339
22345 22346 22354 22357 22364 22367 22375 22376 22382 22389 22393
22398
22417 22435 22453 22471 22489 22496 22499 22501 22508 22512 22515
22521
22526 22533 22534 22543 22544 22551 22556 22565 22578 22587 22619
22637
22655 22673 22691 22714 22736 22741 22758 22763 22785 22839 22857
22875 22893 22916 22938 22961 22983 23008 23012
23015...
_______________________
Waow, thanks again, Jongware!
(this sequence is now in the OEIS as A124177)
Best,
É.
