Digit wheel
We will use the hereunder Digit wheel to illustrate our iteration idea of the day
To go (on the wheel) from 1 to 2 we have to travel a single arc which is produced by an angle of 36 degrees.
We have thus 12 —> 36 (which is the result of 1 arc x 36 degrees)
To go (on the wheel) from 3 to 6 we have to travel 3 arcs which are produced by an angle of 108 degrees.
We have thus 12 —> 36 —> 108 (the result of 3 x 36 degrees)
To go (on the wheel) from 1 to 0 we have to travel 9 arcs (we always travel clockwise) and to go from 0 to 8 we have to travel 8 arcs, for a total of 9 + 8 = 17 arcs or 612 degrees.
We have thus 12 —> 36 —> 108 —> 612 (the result of 17 x 36 degrees)
To go from 6 to 1 we have to travel 5 arcs and to go from 1 to 2 we have to travel 1 arc, for a total of 5 + 1 = 6 arcs or 216 degrees.
We have thus 12 —> 36 —> 108 —> 612 —> 216 (the result of 6 x 36)
To go from 2 to 1 (clockwise) we have to travel 9 arcs and to go from 1 to 6 we have to travel 5 arcs, for a total of 9 + 5 = 14 arcs or 504 degrees.
We have thus 12 —> 36 —> 108 —> 612 —> 216 —> 504 (the result of 14 x 36)
. . .
« 12 » ends in a loop (corrected by JMF, see below):
12 —> 36 —> 108 —> 612 —> 216 —> 504 —> 324 —> 396 —> 468, 144, 468
We could start a «successive iterations» table with a(1) = 10:
10, 324, 396, 468, 144, 468 (loop)
11, 360, 252, 360 (loop)
12, 36, 108, 612, 216, 504, 324, 396, 468, 144, 468 (loop)
13, 72, 180, 324, 396, 468, 144, 468 (loop)
14, 108, 612, 216, 504, 324, 396, 468, 144, 468 (loop)
15, 144, 468, 144 (loop)
16, 180, 324, 396, 468, 144, 468 (loop)
17, 216, 504, 324, 396, 468, 144, 468 (loop)
18, 252, 360, 252 (loop)
19, 288, 576, 324, 396, 468, 144, 468 (loop)
20, 288, 576, 324, 396, 468, 144, 468 (loop)
21, 324, 396, 468, 144, 468 (loop)
22, 360, 252, 360 (loop)
23, 36, 108, 612, 216, 504, 324, 396, 468, 144, 468 (loop)
. . .
We quickly understand that all integers < 100 loop.
Is there a hope to escape loops for a(n)s > 100?
____________________
Update #1, August 7th, 2024
Jean-Marc Falcoz was quick to notice this:
JMF
J’ai une différence avec tes termes :
j’ai {12, 36, 108, 612, 216, 504, 324, 396, 468, 144, 468,...}
au lieu de tes 12 —> 36 —> 108 —> 612 —> 216 —> 504 —> 324 —> 396 —> 576 —> 324.
Geee, you are right, Jean-Marc, sorry... I’ve corrected the above table, thank you!
____________________
Update#2, August 8th, 2024
Jean-Marc Falcoz is sure (like me now) that all iterations, whatever the starting integer, will end in a loop.
Google translation (in blue) immediately after his French quote.
[Jean-Marc’s hereunder list suggests that 540 is the unique fixed point of the iteration; indeed 540 —> 540, the result of (9 + 6) x 36 degrees].
JMF
> Sauf erreur de ma part, ça boucle forcément.
Ci-dessous, les termes pour n=12 à n=1000, et tout en bas, 200 suites, avec pour premier terme un grand nombre aléatoire.
Très en gros, en partant de 10^N, on aura au maximum 36*10*N comme première étape, ce qui est bien plus petit, quel que soit le N de départ. Au bout de quelques itérations (= diminution du nombre), on aboutit à un nombre de 3 chiffres qui est multiple de 36, et l’image suivante ne peut pas avoir 4 chiffres, donc ça boucle obligatoirement.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
> Unless I’m mistaken, it necessarily loops.
Below, the terms for n=12 to n=1000, and at the very bottom, 200 sequences, with a large random number as the first term.
Very roughly, starting from 10^N, we will have at most 36*10*N as the first step, which is much smaller, whatever the starting N. After a few iterations (= decrease in the number), we end up with a 3-digit number that is a multiple of 36, and the next image cannot have 4 digits, so it necessarily loops.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
12 : {12,36,108,612,216,504,324,396,468,144,468}
13 : {13,72,180,324,396,468,144,468}
14 : {14,108,612,216,504,324,396,468,144,468}
15 : {15,144,468,144}
16 : {16,180,324,396,468,144,468}
17 : {17,216,504,324,396,468,144,468}
18 : {18,252,360,252}
19 : {19,288,576,396,468,144,468}
20 : {20,288,576,396,468,144,468}
21 : {21,324,396,468,144,468}
22 : {22,360,252,360}
23 : {23,36,108,612,216,504,324,396,468,144,468}
24 : {24,72,180,324,396,468,144,468}
25 : {25,108,612,216,504,324,396,468,144,468}
26 : {26,144,468,144}
27 : {27,180,324,396,468,144,468}
28 : {28,216,504,324,396,468,144,468}
29 : {29,252,360,252}
30 : {30,252,360,252}
31 : {31,288,576,396,468,144,468}
32 : {32,324,396,468,144,468}
33 : {33,360,252,360}
34 : {34,36,108,612,216,504,324,396,468,144,468}
35 : {35,72,180,324,396,468,144,468}
36 : {36,108,612,216,504,324,396,468,144,468}
37 : {37,144,468,144}
38 : {38,180,324,396,468,144,468}
39 : {39,216,504,324,396,468,144,468}
40 : {40,216,504,324,396,468,144,468}
41 : {41,252,360,252}
42 : {42,288,576,396,468,144,468}
43 : {43,324,396,468,144,468}
44 : {44,360,252,360}
45 : {45,36,108,612,216,504,324,396,468,144,468}
46 : {46,72,180,324,396,468,144,468}
47 : {47,108,612,216,504,324,396,468,144,468}
48 : {48,144,468,144}
49 : {49,180,324,396,468,144,468}
50 : {50,180,324,396,468,144,468}
51 : {51,216,504,324,396,468,144,468}
52 : {52,252,360,252}
53 : {53,288,576,396,468,144,468}
54 : {54,324,396,468,144,468}
55 : {55,360,252,360}
56 : {56,36,108,612,216,504,324,396,468,144,468}
57 : {57,72,180,324,396,468,144,468}
58 : {58,108,612,216,504,324,396,468,144,468}
59 : {59,144,468,144}
60 : {60,144,468,144}
61 : {61,180,324,396,468,144,468}
62 : {62,216,504,324,396,468,144,468}
63 : {63,252,360,252}
64 : {64,288,576,396,468,144,468}
65 : {65,324,396,468,144,468}
66 : {66,360,252,360}
67 : {67,36,108,612,216,504,324,396,468,144,468}
68 : {68,72,180,324,396,468,144,468}
69 : {69,108,612,216,504,324,396,468,144,468}
70 : {70,108,612,216,504,324,396,468,144,468}
71 : {71,144,468,144}
72 : {72,180,324,396,468,144,468}
73 : {73,216,504,324,396,468,144,468}
74 : {74,252,360,252}
75 : {75,288,576,396,468,144,468}
76 : {76,324,396,468,144,468}
77 : {77,360,252,360}
78 : {78,36,108,612,216,504,324,396,468,144,468}
79 : {79,72,180,324,396,468,144,468}
80 : {80,72,180,324,396,468,144,468}
81 : {81,108,612,216,504,324,396,468,144,468}
82 : {82,144,468,144}
83 : {83,180,324,396,468,144,468}
84 : {84,216,504,324,396,468,144,468}
85 : {85,252,360,252}
86 : {86,288,576,396,468,144,468}
87 : {87,324,396,468,144,468}
88 : {88,360,252,360}
89 : {89,36,108,612,216,504,324,396,468,144,468}
90 : {90,36,108,612,216,504,324,396,468,144,468}
91 : {91,72,180,324,396,468,144,468}
92 : {92,108,612,216,504,324,396,468,144,468}
93 : {93,144,468,144}
94 : {94,180,324,396,468,144,468}
95 : {95,216,504,324,396,468,144,468}
96 : {96,252,360,252}
97 : {97,288,576,396,468,144,468}
98 : {98,324,396,468,144,468}
99 : {99,360,252,360}
100 : {100,684,288,576,396,468,144,468}
101 : {101,360,252,360}
102 : {102,396,468,144,468}
103 : {103,432,648,432}
104 : {104,468,144,468}
105 : {105,504,324,396,468,144,468}
106 : {106,540,540}
107 : {107,576,396,468,144,468}
108 : {108,612,216,504,324,396,468,144,468}
109 : {109,648,432,648}
110 : {110,684,288,576,396,468,144,468}
111 : {111,720,468,144,468}
112 : {112,396,468,144,468}
113 : {113,432,648,432}
114 : {114,468,144,468}
115 : {115,504,324,396,468,144,468}
116 : {116,540,540}
117 : {117,576,396,468,144,468}
118 : {118,612,216,504,324,396,468,144,468}
119 : {119,648,432,648}
120 : {120,324,396,468,144,468}
121 : {121,360,252,360}
122 : {122,396,468,144,468}
123 : {123,72,180,324,396,468,144,468}
124 : {124,108,612,216,504,324,396,468,144,468}
125 : {125,144,468,144}
126 : {126,180,324,396,468,144,468}
127 : {127,216,504,324,396,468,144,468}
128 : {128,252,360,252}
129 : {129,288,576,396,468,144,468}
130 : {130,324,396,468,144,468}
131 : {131,360,252,360}
132 : {132,396,468,144,468}
133 : {133,432,648,432}
134 : {134,108,612,216,504,324,396,468,144,468}
135 : {135,144,468,144}
136 : {136,180,324,396,468,144,468}
137 : {137,216,504,324,396,468,144,468}
138 : {138,252,360,252}
139 : {139,288,576,396,468,144,468}
140 : {140,324,396,468,144,468}
141 : {141,360,252,360}
142 : {142,396,468,144,468}
143 : {143,432,648,432}
144 : {144,468,144}
145 : {145,144,468,144}
146 : {146,180,324,396,468,144,468}
147 : {147,216,504,324,396,468,144,468}
148 : {148,252,360,252}
149 : {149,288,576,396,468,144,468}
150 : {150,324,396,468,144,468}
151 : {151,360,252,360}
152 : {152,396,468,144,468}
153 : {153,432,648,432}
154 : {154,468,144,468}
155 : {155,504,324,396,468,144,468}
156 : {156,180,324,396,468,144,468}
157 : {157,216,504,324,396,468,144,468}
158 : {158,252,360,252}
159 : {159,288,576,396,468,144,468}
160 : {160,324,396,468,144,468}
161 : {161,360,252,360}
162 : {162,396,468,144,468}
163 : {163,432,648,432}
164 : {164,468,144,468}
165 : {165,504,324,396,468,144,468}
166 : {166,540,540}
167 : {167,216,504,324,396,468,144,468}
168 : {168,252,360,252}
169 : {169,288,576,396,468,144,468}
170 : {170,324,396,468,144,468}
171 : {171,360,252,360}
172 : {172,396,468,144,468}
173 : {173,432,648,432}
174 : {174,468,144,468}
175 : {175,504,324,396,468,144,468}
176 : {176,540,540}
177 : {177,576,396,468,144,468}
178 : {178,252,360,252}
179 : {179,288,576,396,468,144,468}
180 : {180,324,396,468,144,468}
181 : {181,360,252,360}
182 : {182,396,468,144,468}
183 : {183,432,648,432}
184 : {184,468,144,468}
185 : {185,504,324,396,468,144,468}
186 : {186,540,540}
187 : {187,576,396,468,144,468}
188 : {188,612,216,504,324,396,468,144,468}
189 : {189,288,576,396,468,144,468}
190 : {190,324,396,468,144,468}
191 : {191,360,252,360}
192 : {192,396,468,144,468}
193 : {193,432,648,432}
194 : {194,468,144,468}
195 : {195,504,324,396,468,144,468}
196 : {196,540,540}
197 : {197,576,396,468,144,468}
198 : {198,612,216,504,324,396,468,144,468}
199 : {199,648,432,648}
200 : {200,648,432,648}
201 : {201,324,396,468,144,468}
202 : {202,360,252,360}
203 : {203,396,468,144,468}
204 : {204,432,648,432}
205 : {205,468,144,468}
206 : {206,504,324,396,468,144,468}
207 : {207,540,540}
208 : {208,576,396,468,144,468}
209 : {209,612,216,504,324,396,468,144,468}
210 : {210,648,432,648}
211 : {211,684,288,576,396,468,144,468}
212 : {212,360,252,360}
213 : {213,396,468,144,468}
214 : {214,432,648,432}
215 : {215,468,144,468}
216 : {216,504,324,396,468,144,468}
217 : {217,540,540}
218 : {218,576,396,468,144,468}
219 : {219,612,216,504,324,396,468,144,468}
220 : {220,648,432,648}
221 : {221,684,288,576,396,468,144,468}
222 : {222,720,468,144,468}
223 : {223,396,468,144,468}
224 : {224,432,648,432}
225 : {225,468,144,468}
226 : {226,504,324,396,468,144,468}
227 : {227,540,540}
228 : {228,576,396,468,144,468}
229 : {229,612,216,504,324,396,468,144,468}
230 : {230,288,576,396,468,144,468}
231 : {231,324,396,468,144,468}
232 : {232,360,252,360}
233 : {233,396,468,144,468}
234 : {234,72,180,324,396,468,144,468}
235 : {235,108,612,216,504,324,396,468,144,468}
236 : {236,144,468,144}
237 : {237,180,324,396,468,144,468}
238 : {238,216,504,324,396,468,144,468}
239 : {239,252,360,252}
240 : {240,288,576,396,468,144,468}
241 : {241,324,396,468,144,468}
242 : {242,360,252,360}
243 : {243,396,468,144,468}
244 : {244,432,648,432}
245 : {245,108,612,216,504,324,396,468,144,468}
246 : {246,144,468,144}
247 : {247,180,324,396,468,144,468}
248 : {248,216,504,324,396,468,144,468}
249 : {249,252,360,252}
250 : {250,288,576,396,468,144,468}
251 : {251,324,396,468,144,468}
252 : {252,360,252}
253 : {253,396,468,144,468}
254 : {254,432,648,432}
255 : {255,468,144,468}
256 : {256,144,468,144}
257 : {257,180,324,396,468,144,468}
258 : {258,216,504,324,396,468,144,468}
259 : {259,252,360,252}
260 : {260,288,576,396,468,144,468}
261 : {261,324,396,468,144,468}
262 : {262,360,252,360}
263 : {263,396,468,144,468}
264 : {264,432,648,432}
265 : {265,468,144,468}
266 : {266,504,324,396,468,144,468}
267 : {267,180,324,396,468,144,468}
268 : {268,216,504,324,396,468,144,468}
269 : {269,252,360,252}
270 : {270,288,576,396,468,144,468}
271 : {271,324,396,468,144,468}
272 : {272,360,252,360}
273 : {273,396,468,144,468}
274 : {274,432,648,432}
275 : {275,468,144,468}
276 : {276,504,324,396,468,144,468}
277 : {277,540,540}
278 : {278,216,504,324,396,468,144,468}
279 : {279,252,360,252}
280 : {280,288,576,396,468,144,468}
281 : {281,324,396,468,144,468}
282 : {282,360,252,360}
283 : {283,396,468,144,468}
284 : {284,432,648,432}
285 : {285,468,144,468}
286 : {286,504,324,396,468,144,468}
287 : {287,540,540}
288 : {288,576,396,468,144,468}
289 : {289,252,360,252}
290 : {290,288,576,396,468,144,468}
291 : {291,324,396,468,144,468}
292 : {292,360,252,360}
293 : {293,396,468,144,468}
294 : {294,432,648,432}
295 : {295,468,144,468}
296 : {296,504,324,396,468,144,468}
297 : {297,540,540}
298 : {298,576,396,468,144,468}
299 : {299,612,216,504,324,396,468,144,468}
300 : {300,612,216,504,324,396,468,144,468}
301 : {301,288,576,396,468,144,468}
302 : {302,324,396,468,144,468}
303 : {303,360,252,360}
304 : {304,396,468,144,468}
305 : {305,432,648,432}
306 : {306,468,144,468}
307 : {307,504,324,396,468,144,468}
308 : {308,540,540}
309 : {309,576,396,468,144,468}
310 : {310,612,216,504,324,396,468,144,468}
311 : {311,648,432,648}
312 : {312,324,396,468,144,468}
313 : {313,360,252,360}
314 : {314,396,468,144,468}
315 : {315,432,648,432}
316 : {316,468,144,468}
317 : {317,504,324,396,468,144,468}
318 : {318,540,540}
319 : {319,576,396,468,144,468}
320 : {320,612,216,504,324,396,468,144,468}
321 : {321,648,432,648}
322 : {322,684,288,576,396,468,144,468}
323 : {323,360,252,360}
324 : {324,396,468,144,468}
325 : {325,432,648,432}
326 : {326,468,144,468}
327 : {327,504,324,396,468,144,468}
328 : {328,540,540}
329 : {329,576,396,468,144,468}
330 : {330,612,216,504,324,396,468,144,468}
331 : {331,648,432,648}
332 : {332,684,288,576,396,468,144,468}
333 : {333,720,468,144,468}
334 : {334,396,468,144,468}
335 : {335,432,648,432}
336 : {336,468,144,468}
337 : {337,504,324,396,468,144,468}
338 : {338,540,540}
339 : {339,576,396,468,144,468}
340 : {340,252,360,252}
341 : {341,288,576,396,468,144,468}
342 : {342,324,396,468,144,468}
343 : {343,360,252,360}
344 : {344,396,468,144,468}
345 : {345,72,180,324,396,468,144,468}
346 : {346,108,612,216,504,324,396,468,144,468}
347 : {347,144,468,144}
348 : {348,180,324,396,468,144,468}
349 : {349,216,504,324,396,468,144,468}
350 : {350,252,360,252}
351 : {351,288,576,396,468,144,468}
352 : {352,324,396,468,144,468}
353 : {353,360,252,360}
354 : {354,396,468,144,468}
355 : {355,432,648,432}
356 : {356,108,612,216,504,324,396,468,144,468}
357 : {357,144,468,144}
358 : {358,180,324,396,468,144,468}
359 : {359,216,504,324,396,468,144,468}
360 : {360,252,360}
361 : {361,288,576,396,468,144,468}
362 : {362,324,396,468,144,468}
363 : {363,360,252,360}
364 : {364,396,468,144,468}
365 : {365,432,648,432}
366 : {366,468,144,468}
367 : {367,144,468,144}
368 : {368,180,324,396,468,144,468}
369 : {369,216,504,324,396,468,144,468}
370 : {370,252,360,252}
371 : {371,288,576,396,468,144,468}
372 : {372,324,396,468,144,468}
373 : {373,360,252,360}
374 : {374,396,468,144,468}
375 : {375,432,648,432}
376 : {376,468,144,468}
377 : {377,504,324,396,468,144,468}
378 : {378,180,324,396,468,144,468}
379 : {379,216,504,324,396,468,144,468}
380 : {380,252,360,252}
381 : {381,288,576,396,468,144,468}
382 : {382,324,396,468,144,468}
383 : {383,360,252,360}
384 : {384,396,468,144,468}
385 : {385,432,648,432}
386 : {386,468,144,468}
387 : {387,504,324,396,468,144,468}
388 : {388,540,540}
389 : {389,216,504,324,396,468,144,468}
390 : {390,252,360,252}
391 : {391,288,576,396,468,144,468}
392 : {392,324,396,468,144,468}
393 : {393,360,252,360}
394 : {394,396,468,144,468}
395 : {395,432,648,432}
396 : {396,468,144,468}
397 : {397,504,324,396,468,144,468}
398 : {398,540,540}
399 : {399,576,396,468,144,468}
400 : {400,576,396,468,144,468}
401 : {401,252,360,252}
402 : {402,288,576,396,468,144,468}
403 : {403,324,396,468,144,468}
404 : {404,360,252,360}
405 : {405,396,468,144,468}
406 : {406,432,648,432}
407 : {407,468,144,468}
408 : {408,504,324,396,468,144,468}
409 : {409,540,540}
410 : {410,576,396,468,144,468}
411 : {411,612,216,504,324,396,468,144,468}
412 : {412,288,576,396,468,144,468}
413 : {413,324,396,468,144,468}
414 : {414,360,252,360}
415 : {415,396,468,144,468}
416 : {416,432,648,432}
417 : {417,468,144,468}
418 : {418,504,324,396,468,144,468}
419 : {419,540,540}
420 : {420,576,396,468,144,468}
421 : {421,612,216,504,324,396,468,144,468}
422 : {422,648,432,648}
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____________________
Good job, Jean-Marc, many thanks!
I think I will re-use this wheel later – dunno how yet!
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