Start at $y=1$, then start observing sequences $a_{(x,y)}(n)$ for $x=1,2,3,\dots$ and stop once you reach a sequence $a(n)$ that is exactly the same as one of the previously observed sequences. This $x$ is the stopping point for this $y$. Now move onto the next natural number $y$, and repeat, considering all sequences so far, to find the stopping point for that $y$.
If the sequences are defined as the following:
$$ a_{(x,y)}(n) = (n^x+1)\bmod y$$
How to directly find the stopping $x$ for some $y$?
Edit: Examples of the process:
a_(1,1)= 0, 0,...
a_(2,1)= 0, 0,...
For y=1, we stop at x=2, since a(1,1)=a(2,1)
a_(1,2)= 1, 0, 1,...
a_(2,2)= 1, 0, 1,...
For y=2, we stop at x=2, since a(1,2)=a(2,2)
a_(1,3)= 1, 2, 0, 1,...
a_(2,3)= 1, 2, 2, 1,...
a_(3,3)= 1, 2, 0, 1,...
For y=3, we stop at x=3, since a(1,3)=a(3,3)
a_(1,4)= 1, 2, 3, 0, 1,...
a_(2,4)= 1, 2, 1, 2, 1,...
a_(3,4)= 1, 2, 1, 0, 1,...
a_(4,4)= 1, 2, 1, 2, 1,...
For y=4, we stop at x=4, since a(2,4)=a(4,4)
a_(1,5)= 1, 2, 3, 4, 0, 1,...
a_(2,5)= 1, 2, 0, 0, 2, 1,...
a_(3,5)= 1, 2, 4, 3, 0, 1,...
a_(4,5)= 1, 2, 2, 2, 2, 1,...
a_(5,5)= 1, 2, 3, 4, 0, 1,...
For y=5, we stop at x=5, since a(1,5)=a(5,5)
a_(1,6)= 1, 2, 3, 4, 5, 0, 1,...
a_(2,6)= 1, 2, 5, 4, 5, 2, 1,...
a_(3,6)= 1, 2, 3, 4, 5, 0, 1,...
For y=6, we stop at x=3, since a(1,6)=a(3,6)
I'm not familiar with modular arithmetic, so I tried to compute the results and observe if I can see any patterns among them.
I've observed that if $y$ is a prime number, then the stopping point is exactly $x=y$ .
I'm not sure how to find a general way to calculate/compute the stopping $x$ if $y$ is composite.
Here are the computed results: y x [prime factorization of y]
1 2 []
2 2 [2]
3 3 [3]
4 4 [2, 2]
5 5 [5]
6 3 [2, 3]
7 7 [7]
8 5 [2, 2, 2]
9 8 [3, 3]
10 5 [2, 5]
11 11 [11]
12 4 [2, 2, 3]
13 13 [13]
14 7 [2, 7]
15 5 [3, 5]
16 8 [2, 2, 2, 2]
17 17 [17]
18 8 [2, 3, 3]
19 19 [19]
20 6 [2, 2, 5]
21 7 [3, 7]
22 11 [2, 11]
23 23 [23]
24 5 [2, 2, 2, 3]
25 22 [5, 5]
26 13 [2, 13]
27 21 [3, 3, 3]
28 8 [2, 2, 7]
29 29 [29]
30 5 [2, 3, 5]
31 31 [31]
32 13 [2, 2, 2, 2, 2]
33 11 [3, 11]
34 17 [2, 17]
35 13 [5, 7]
36 8 [2, 2, 3, 3]
37 37 [37]
38 19 [2, 19]
39 13 [3, 13]
40 7 [2, 2, 2, 5]
41 41 [41]
42 7 [2, 3, 7]
43 43 [43]
44 12 [2, 2, 11]
45 14 [3, 3, 5]
46 23 [2, 23]
47 47 [47]
48 8 [2, 2, 2, 2, 3]
49 44 [7, 7]
50 22 [2, 5, 5]
51 17 [3, 17]
52 14 [2, 2, 13]
53 53 [53]
54 21 [2, 3, 3, 3]
55 21 [5, 11]
56 9 [2, 2, 2, 7]
57 19 [3, 19]
58 29 [2, 29]
59 59 [59]
60 6 [2, 2, 3, 5]
61 61 [61]
62 31 [2, 31]
63 8 [3, 3, 7]
64 22 [2, 2, 2, 2, 2, 2]
65 13 [5, 13]
66 11 [2, 3, 11]
67 67 [67]
68 18 [2, 2, 17]
69 23 [3, 23]
70 13 [2, 5, 7]
71 71 [71]
72 9 [2, 2, 2, 3, 3]
73 73 [73]
74 37 [2, 37]
75 22 [3, 5, 5]
76 20 [2, 2, 19]
77 31 [7, 11]
78 13 [2, 3, 13]
79 79 [79]
80 8 [2, 2, 2, 2, 5]
81 58 [3, 3, 3, 3]
82 41 [2, 41]
83 83 [83]
84 8 [2, 2, 3, 7]
85 17 [5, 17]
86 43 [2, 43]
87 29 [3, 29]
88 13 [2, 2, 2, 11]
89 89 [89]
90 14 [2, 3, 3, 5]
91 13 [7, 13]
92 24 [2, 2, 23]
93 31 [3, 31]
94 47 [2, 47]
95 37 [5, 19]
96 13 [2, 2, 2, 2, 2, 3]
97 97 [97]
98 44 [2, 7, 7]
99 32 [3, 3, 11]
100 22 [2, 2, 5, 5]
101 101 [101]
102 17 [2, 3, 17]
103 103 [103]
104 15 [2, 2, 2, 13]
105 13 [3, 5, 7]
106 53 [2, 53]
107 107 [107]
108 21 [2, 2, 3, 3, 3]
109 109 [109]
110 21 [2, 5, 11]
111 37 [3, 37]
112 16 [2, 2, 2, 2, 7]
113 113 [113]
114 19 [2, 3, 19]
115 45 [5, 23]
116 30 [2, 2, 29]
117 14 [3, 3, 13]
118 59 [2, 59]
119 49 [7, 17]
120 7 [2, 2, 2, 3, 5]
121 112 [11, 11]
122 61 [2, 61]
123 41 [3, 41]
124 32 [2, 2, 31]
125 103 [5, 5, 5]
126 8 [2, 3, 3, 7]
127 127 [127]
128 39 [2, 2, 2, 2, 2, 2, 2]
129 43 [3, 43]
130 13 [2, 5, 13]
131 131 [131]
132 12 [2, 2, 3, 11]
133 19 [7, 19]
134 67 [2, 67]
135 39 [3, 3, 3, 5]
136 19 [2, 2, 2, 17]
137 137 [137]
138 23 [2, 3, 23]
139 139 [139]
140 14 [2, 2, 5, 7]
141 47 [3, 47]
142 71 [2, 71]
143 61 [11, 13]
144 16 [2, 2, 2, 2, 3, 3]
145 29 [5, 29]
146 73 [2, 73]
147 44 [3, 7, 7]
148 38 [2, 2, 37]
149 149 [149]
150 22 [2, 3, 5, 5]
151 151 [151]
152 21 [2, 2, 2, 19]
153 50 [3, 3, 17]
154 31 [2, 7, 11]
155 61 [5, 31]
156 14 [2, 2, 3, 13]
157 157 [157]
158 79 [2, 79]
159 53 [3, 53]
160 13 [2, 2, 2, 2, 2, 5]
161 67 [7, 23]
162 58 [2, 3, 3, 3, 3]
163 163 [163]
164 42 [2, 2, 41]
165 21 [3, 5, 11]
166 83 [2, 83]
167 167 [167]
168 9 [2, 2, 2, 3, 7]
169 158 [13, 13]
170 17 [2, 5, 17]
171 20 [3, 3, 19]
172 44 [2, 2, 43]
173 173 [173]
174 29 [2, 3, 29]
175 62 [5, 5, 7]
176 24 [2, 2, 2, 2, 11]
177 59 [3, 59]
178 89 [2, 89]
179 179 [179]
180 14 [2, 2, 3, 3, 5]
181 181 [181]
182 13 [2, 7, 13]
183 61 [3, 61]
184 25 [2, 2, 2, 23]
185 37 [5, 37]
186 31 [2, 3, 31]
187 81 [11, 17]
188 48 [2, 2, 47]
189 21 [3, 3, 3, 7]
190 37 [2, 5, 19]
191 191 [191]
192 22 [2, 2, 2, 2, 2, 2, 3]
193 193 [193]
194 97 [2, 97]
195 13 [3, 5, 13]
196 44 [2, 2, 7, 7]
197 197 [197]
198 32 [2, 3, 3, 11]
199 199 [199]
200 23 [2, 2, 2, 5, 5]
201 67 [3, 67]
202 101 [2, 101]
203 85 [7, 29]
204 18 [2, 2, 3, 17]
205 41 [5, 41]
206 103 [2, 103]
207 68 [3, 3, 23]
208 16 [2, 2, 2, 2, 13]
209 91 [11, 19]
210 13 [2, 3, 5, 7]
211 211 [211]
212 54 [2, 2, 53]
213 71 [3, 71]
214 107 [2, 107]
215 85 [5, 43]
216 21 [2, 2, 2, 3, 3, 3]
217 31 [7, 31]
218 109 [2, 109]
219 73 [3, 73]
220 22 [2, 2, 5, 11]
221 49 [13, 17]
222 37 [2, 3, 37]
223 223 [223]
224 29 [2, 2, 2, 2, 2, 7]
225 62 [3, 3, 5, 5]
226 113 [2, 113]
227 227 [227]
228 20 [2, 2, 3, 19]
229 229 [229]
230 45 [2, 5, 23]
231 31 [3, 7, 11]
232 31 [2, 2, 2, 29]
233 233 [233]
234 14 [2, 3, 3, 13]
235 93 [5, 47]
236 60 [2, 2, 59]
237 79 [3, 79]
238 49 [2, 7, 17]
239 239 [239]
240 8 [2, 2, 2, 2, 3, 5]
241 241 [241]
242 112 [2, 11, 11]
243 167 [3, 3, 3, 3, 3]
244 62 [2, 2, 61]
245 86 [5, 7, 7]
246 41 [2, 3, 41]
247 37 [13, 19]
248 33 [2, 2, 2, 31]
249 83 [3, 83]
250 103 [2, 5, 5, 5]
251 251 [251]
252 8 [2, 2, 3, 3, 7]
253 111 [11, 23]
254 127 [2, 127]
255 17 [3, 5, 17]
256 72 [2, 2, 2, 2, 2, 2, 2, 2]
257 257 [257]
258 43 [2, 3, 43]
259 37 [7, 37]
260 14 [2, 2, 5, 13]
261 86 [3, 3, 29]
262 131 [2, 131]
263 263 [263]
264 13 [2, 2, 2, 3, 11]
265 53 [5, 53]
266 19 [2, 7, 19]
267 89 [3, 89]
268 68 [2, 2, 67]
269 269 [269]
270 39 [2, 3, 3, 3, 5]
271 271 [271]
272 20 [2, 2, 2, 2, 17]
273 13 [3, 7, 13]
274 137 [2, 137]
275 22 [5, 5, 11]
276 24 [2, 2, 3, 23]
277 277 [277]
278 139 [2, 139]
279 32 [3, 3, 31]
280 15 [2, 2, 2, 5, 7]
281 281 [281]
282 47 [2, 3, 47]
283 283 [283]
284 72 [2, 2, 71]
285 37 [3, 5, 19]
286 61 [2, 11, 13]
287 121 [7, 41]
288 29 [2, 2, 2, 2, 2, 3, 3]
289 274 [17, 17]
290 29 [2, 5, 29]
291 97 [3, 97]
292 74 [2, 2, 73]
293 293 [293]
294 44 [2, 3, 7, 7]
295 117 [5, 59]
296 39 [2, 2, 2, 37]
297 93 [3, 3, 3, 11]
298 149 [2, 149]
299 133 [13, 23]
300 22 [2, 2, 3, 5, 5]
301 43 [7, 43]
302 151 [2, 151]
303 101 [3, 101]
304 40 [2, 2, 2, 2, 19]
305 61 [5, 61]
306 50 [2, 3, 3, 17]
307 307 [307]
308 32 [2, 2, 7, 11]
309 103 [3, 103]
310 61 [2, 5, 31]
311 311 [311]
312 15 [2, 2, 2, 3, 13]
313 313 [313]
314 157 [2, 157]
315 14 [3, 3, 5, 7]
316 80 [2, 2, 79]
317 317 [317]
318 53 [2, 3, 53]
319 141 [11, 29]
320 22 [2, 2, 2, 2, 2, 2, 5]
321 107 [3, 107]
322 67 [2, 7, 23]
323 145 [17, 19]
324 58 [2, 2, 3, 3, 3, 3]
325 62 [5, 5, 13]
326 163 [2, 163]
327 109 [3, 109]
328 43 [2, 2, 2, 41]
329 139 [7, 47]
330 21 [2, 3, 5, 11]
331 331 [331]
332 84 [2, 2, 83]
333 38 [3, 3, 37]
334 167 [2, 167]
335 133 [5, 67]
336 16 [2, 2, 2, 2, 3, 7]
337 337 [337]
338 158 [2, 13, 13]
339 113 [3, 113]
340 18 [2, 2, 5, 17]
341 31 [11, 31]
342 20 [2, 3, 3, 19]
343 297 [7, 7, 7]
344 45 [2, 2, 2, 43]
345 45 [3, 5, 23]
346 173 [2, 173]
347 347 [347]
348 30 [2, 2, 3, 29]
349 349 [349]
350 62 [2, 5, 5, 7]
351 39 [3, 3, 3, 13]
352 45 [2, 2, 2, 2, 2, 11]
353 353 [353]
354 59 [2, 3, 59]
355 141 [5, 71]
356 90 [2, 2, 89]
357 49 [3, 7, 17]
358 179 [2, 179]
359 359 [359]
360 15 [2, 2, 2, 3, 3, 5]
361 344 [19, 19]
362 181 [2, 181]
363 112 [3, 11, 11]
364 14 [2, 2, 7, 13]
365 73 [5, 73]
366 61 [2, 3, 61]
367 367 [367]
368 48 [2, 2, 2, 2, 23]
369 122 [3, 3, 41]
370 37 [2, 5, 37]
371 157 [7, 53]
372 32 [2, 2, 3, 31]
373 373 [373]
374 81 [2, 11, 17]
375 103 [3, 5, 5, 5]
376 49 [2, 2, 2, 47]
377 85 [13, 29]
378 21 [2, 3, 3, 3, 7]
379 379 [379]
380 38 [2, 2, 5, 19]
381 127 [3, 127]
382 191 [2, 191]
383 383 [383]
384 39 [2, 2, 2, 2, 2, 2, 2, 3]
385 61 [5, 7, 11]
386 193 [2, 193]
387 44 [3, 3, 43]
388 98 [2, 2, 97]
389 389 [389]
390 13 [2, 3, 5, 13]
391 177 [17, 23]
392 45 [2, 2, 2, 7, 7]
393 131 [3, 131]
394 197 [2, 197]
395 157 [5, 79]
396 32 [2, 2, 3, 3, 11]
397 397 [397]
398 199 [2, 199]
399 19 [3, 7, 19]
400 24 [2, 2, 2, 2, 5, 5]
401 401 [401]
402 67 [2, 3, 67]
403 61 [13, 31]
404 102 [2, 2, 101]
405 112 [3, 3, 3, 3, 5]
406 85 [2, 7, 29]
407 181 [11, 37]
408 19 [2, 2, 2, 3, 17]
409 409 [409]
410 41 [2, 5, 41]
411 137 [3, 137]
412 104 [2, 2, 103]
413 175 [7, 59]
414 68 [2, 3, 3, 23]
415 165 [5, 83]
416 29 [2, 2, 2, 2, 2, 13]
417 139 [3, 139]
418 91 [2, 11, 19]
419 419 [419]
420 14 [2, 2, 3, 5, 7]
421 421 [421]
422 211 [2, 211]
423 140 [3, 3, 47]
424 55 [2, 2, 2, 53]
425 82 [5, 5, 17]
426 71 [2, 3, 71]
427 61 [7, 61]
428 108 [2, 2, 107]
429 61 [3, 11, 13]
430 85 [2, 5, 43]
431 431 [431]
432 40 [2, 2, 2, 2, 3, 3, 3]
433 433 [433]
434 31 [2, 7, 31]
435 29 [3, 5, 29]
436 110 [2, 2, 109]
437 199 [19, 23]
438 73 [2, 3, 73]
439 439 [439]
440 23 [2, 2, 2, 5, 11]
441 44 [3, 3, 7, 7]
442 49 [2, 13, 17]
443 443 [443]
444 38 [2, 2, 3, 37]
445 89 [5, 89]
446 223 [2, 223]
447 149 [3, 149]
448 54 [2, 2, 2, 2, 2, 2, 7]
449 449 [449]
450 62 [2, 3, 3, 5, 5]
451 41 [11, 41]
452 114 [2, 2, 113]
453 151 [3, 151]
454 227 [2, 227]
455 13 [5, 7, 13]
456 21 [2, 2, 2, 3, 19]
457 457 [457]
458 229 [2, 229]
459 147 [3, 3, 3, 17]
460 46 [2, 2, 5, 23]
461 461 [461]
462 31 [2, 3, 7, 11]
463 463 [463]
464 32 [2, 2, 2, 2, 29]
465 61 [3, 5, 31]
466 233 [2, 233]
467 467 [467]
468 14 [2, 2, 3, 3, 13]
469 67 [7, 67]
470 93 [2, 5, 47]
471 157 [3, 157]
472 61 [2, 2, 2, 59]
473 211 [11, 43]
474 79 [2, 3, 79]
475 182 [5, 5, 19]
476 50 [2, 2, 7, 17]
477 158 [3, 3, 53]
478 239 [2, 239]
479 479 [479]
480 13 [2, 2, 2, 2, 2, 3, 5]
481 37 [13, 37]
482 241 [2, 241]
483 67 [3, 7, 23]
484 112 [2, 2, 11, 11]
485 97 [5, 97]
486 167 [2, 3, 3, 3, 3, 3]
487 487 [487]
488 63 [2, 2, 2, 61]
489 163 [3, 163]
490 86 [2, 5, 7, 7]
491 491 [491]
492 42 [2, 2, 3, 41]
493 113 [17, 29]
494 37 [2, 13, 19]
495 62 [3, 3, 5, 11]
496 64 [2, 2, 2, 2, 31]
497 211 [7, 71]
498 83 [2, 3, 83]
499 499 [499]
500 103 [2, 2, 5, 5, 5]
501 167 [3, 167]
502 251 [2, 251]
503 503 [503]
504 9 [2, 2, 2, 3, 3, 7]
505 101 [5, 101]
506 111 [2, 11, 23]
507 158 [3, 13, 13]
508 128 [2, 2, 127]
509 509 [509]
510 17 [2, 3, 5, 17]
511 73 [7, 73]
512 137 [2, 2, 2, 2, 2, 2, 2, 2, 2]
513 21 [3, 3, 3, 19]
514 257 [2, 257]
515 205 [5, 103]
516 44 [2, 2, 3, 43]
517 231 [11, 47]
518 37 [2, 7, 37]
519 173 [3, 173]
520 15 [2, 2, 2, 5, 13]
521 521 [521]
522 86 [2, 3, 3, 29]
523 523 [523]
524 132 [2, 2, 131]
525 62 [3, 5, 5, 7]
526 263 [2, 263]
527 241 [17, 31]
528 24 [2, 2, 2, 2, 3, 11]
529 508 [23, 23]
530 53 [2, 5, 53]
531 176 [3, 3, 59]
532 20 [2, 2, 7, 19]
533 121 [13, 41]
534 89 [2, 3, 89]
535 213 [5, 107]
536 69 [2, 2, 2, 67]
537 179 [3, 179]
538 269 [2, 269]
539 212 [7, 7, 11]
540 39 [2, 2, 3, 3, 3, 5]
541 541 [541]
542 271 [2, 271]
543 181 [3, 181]
544 21 [2, 2, 2, 2, 2, 17]
545 109 [5, 109]
546 13 [2, 3, 7, 13]
547 547 [547]
548 138 [2, 2, 137]
549 62 [3, 3, 61]
550 22 [2, 5, 5, 11]
551 253 [19, 29]
552 25 [2, 2, 2, 3, 23]
553 79 [7, 79]
554 277 [2, 277]
555 37 [3, 5, 37]
556 140 [2, 2, 139]
557 557 [557]
558 32 [2, 3, 3, 31]
559 85 [13, 43]
560 16 [2, 2, 2, 2, 5, 7]
561 81 [3, 11, 17]
562 281 [2, 281]
563 563 [563]
564 48 [2, 2, 3, 47]
565 113 [5, 113]
566 283 [2, 283]
567 58 [3, 3, 3, 3, 7]
568 73 [2, 2, 2, 71]
569 569 [569]
570 37 [2, 3, 5, 19]
571 571 [571]
572 62 [2, 2, 11, 13]
573 191 [3, 191]
574 121 [2, 7, 41]
575 222 [5, 5, 23]
576 54 [2, 2, 2, 2, 2, 2, 3, 3]
577 577 [577]
578 274 [2, 17, 17]
579 193 [3, 193]
580 30 [2, 2, 5, 29]
581 247 [7, 83]
582 97 [2, 3, 97]
583 261 [11, 53]
584 75 [2, 2, 2, 73]
585 14 [3, 3, 5, 13]
586 293 [2, 293]
587 587 [587]
588 44 [2, 2, 3, 7, 7]
589 91 [19, 31]
590 117 [2, 5, 59]
591 197 [3, 197]
592 40 [2, 2, 2, 2, 37]
593 593 [593]
594 93 [2, 3, 3, 3, 11]
595 49 [5, 7, 17]
596 150 [2, 2, 149]
597 199 [3, 199]
598 133 [2, 13, 23]
599 599 [599]
600 23 [2, 2, 2, 3, 5, 5]
601 601 [601]
602 43 [2, 7, 43]
603 68 [3, 3, 67]
604 152 [2, 2, 151]
605 222 [5, 11, 11]
606 101 [2, 3, 101]
607 607 [607]
608 77 [2, 2, 2, 2, 2, 19]
609 85 [3, 7, 29]
610 61 [2, 5, 61]
611 277 [13, 47]
612 50 [2, 2, 3, 3, 17]
613 613 [613]
614 307 [2, 307]
615 41 [3, 5, 41]
616 33 [2, 2, 2, 7, 11]
617 617 [617]
618 103 [2, 3, 103]
619 619 [619]
620 62 [2, 2, 5, 31]
621 201 [3, 3, 3, 23]
622 311 [2, 311]
623 265 [7, 89]
624 16 [2, 2, 2, 2, 3, 13]
625 504 [5, 5, 5, 5]
626 313 [2, 313]
627 91 [3, 11, 19]
628 158 [2, 2, 157]
629 145 [17, 37]
630 14 [2, 3, 3, 5, 7]
631 631 [631]
632 81 [2, 2, 2, 79]
633 211 [3, 211]
634 317 [2, 317]
635 253 [5, 127]
636 54 [2, 2, 3, 53]
637 86 [7, 7, 13]
638 141 [2, 11, 29]
639 212 [3, 3, 71]
640 39 [2, 2, 2, 2, 2, 2, 2, 5]
641 641 [641]
642 107 [2, 3, 107]
643 643 [643]
644 68 [2, 2, 7, 23]
645 85 [3, 5, 43]
646 145 [2, 17, 19]
647 647 [647]
648 58 [2, 2, 2, 3, 3, 3, 3]
649 291 [11, 59]
650 62 [2, 5, 5, 13]
651 31 [3, 7, 31]
652 164 [2, 2, 163]
653 653 [653]
654 109 [2, 3, 109]
655 261 [5, 131]
656 44 [2, 2, 2, 2, 41]
657 74 [3, 3, 73]
658 139 [2, 7, 47]
659 659 [659]
660 22 [2, 2, 3, 5, 11]
661 661 [661]
662 331 [2, 331]
663 49 [3, 13, 17]
664 85 [2, 2, 2, 83]
665 37 [5, 7, 19]
666 38 [2, 3, 3, 37]
667 309 [23, 29]
668 168 [2, 2, 167]
669 223 [3, 223]
670 133 [2, 5, 67]
671 61 [11, 61]
672 29 [2, 2, 2, 2, 2, 3, 7]
673 673 [673]
674 337 [2, 337]
675 183 [3, 3, 3, 5, 5]
676 158 [2, 2, 13, 13]
677 677 [677]
678 113 [2, 3, 113]
679 97 [7, 97]
680 19 [2, 2, 2, 5, 17]
681 227 [3, 227]
682 31 [2, 11, 31]
683 683 [683]
684 20 [2, 2, 3, 3, 19]
685 137 [5, 137]
686 297 [2, 7, 7, 7]
687 229 [3, 229]
688 88 [2, 2, 2, 2, 43]
689 157 [13, 53]
690 45 [2, 3, 5, 23]
691 691 [691]
692 174 [2, 2, 173]
693 32 [3, 3, 7, 11]
694 347 [2, 347]
695 277 [5, 139]
696 31 [2, 2, 2, 3, 29]
697 81 [17, 41]
698 349 [2, 349]
699 233 [3, 233]
700 62 [2, 2, 5, 5, 7]
701 701 [701]
702 39 [2, 3, 3, 3, 13]
703 37 [19, 37]
704 86 [2, 2, 2, 2, 2, 2, 11]
705 93 [3, 5, 47]
706 353 [2, 353]
707 301 [7, 101]
708 60 [2, 2, 3, 59]
709 709 [709]
710 141 [2, 5, 71]
711 80 [3, 3, 79]
712 91 [2, 2, 2, 89]
713 331 [23, 31]
714 49 [2, 3, 7, 17]
715 61 [5, 11, 13]
716 180 [2, 2, 179]
717 239 [3, 239]
718 359 [2, 359]
719 719 [719]
720 16 [2, 2, 2, 2, 3, 3, 5]
721 103 [7, 103]
722 344 [2, 19, 19]
723 241 [3, 241]
724 182 [2, 2, 181]
725 142 [5, 5, 29]
726 112 [2, 3, 11, 11]
727 727 [727]
728 15 [2, 2, 2, 7, 13]
729 492 [3, 3, 3, 3, 3, 3]
730 73 [2, 5, 73]
731 337 [17, 43]
732 62 [2, 2, 3, 61]
733 733 [733]
734 367 [2, 367]
735 86 [3, 5, 7, 7]
736 93 [2, 2, 2, 2, 2, 23]
737 331 [11, 67]
738 122 [2, 3, 3, 41]
739 739 [739]
740 38 [2, 2, 5, 37]
741 37 [3, 13, 19]
742 157 [2, 7, 53]
743 743 [743]
744 33 [2, 2, 2, 3, 31]
745 149 [5, 149]
746 373 [2, 373]
747 248 [3, 3, 83]
748 82 [2, 2, 11, 17]
749 319 [7, 107]
750 103 [2, 3, 5, 5, 5]
751 751 [751]
752 96 [2, 2, 2, 2, 47]
753 251 [3, 251]
754 85 [2, 13, 29]
755 301 [5, 151]
756 21 [2, 2, 3, 3, 3, 7]
757 757 [757]
758 379 [2, 379]
759 111 [3, 11, 23]
760 39 [2, 2, 2, 5, 19]
761 761 [761]
762 127 [2, 3, 127]
763 109 [7, 109]
764 192 [2, 2, 191]
765 50 [3, 3, 5, 17]
766 383 [2, 383]
767 349 [13, 59]
768 72 [2, 2, 2, 2, 2, 2, 2, 2, 3]
769 769 [769]
770 61 [2, 5, 7, 11]
771 257 [3, 257]
772 194 [2, 2, 193]
773 773 [773]
774 44 [2, 3, 3, 43]
775 62 [5, 5, 31]
776 99 [2, 2, 2, 97]
777 37 [3, 7, 37]
I've also noticed if the composite number is of form $y=2p$ or $y=3p$, where $p$ is a prime number not equal to $2$ or $3$ respectively; Then the stopping point is exactly $x=p$ .
If we try to extend to the $y=5p$, this only holds if $p=4k+1,k\in\mathbb N$ .
Otherwise, $y=5p$ will have the stopping point equal to $2p-1$ .
How can we extend this to all composite numbers which are a multiple of two primes, and show why are these observations so far correct?
Can we find a general pattern for all composite numbers?
That is, given any $y$, directly find the stopping $x$.