Base 105: Iteration Frequencies: Full Cycle

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789ABCDEF
GHIJKLMNOPQRSTUV
WXYZabcdefghijkl
mnopqrstuvwxyzΑΒ
ΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣ
ΤΥΦΧΨΩαβγδεζηθικ
λμνξοπρςστυφχψωА

Color Key
OrangeFixed Pointnumber that becomes itself after one iteration
GreenFinal Numbersall numbers eventually reached, from all cycles
PurpleCycle Startiterations include entry into cycle, assuming it repeats
RedFull Cycleiterations include entire cycle, proving it repeats
BlueStart Numberssubset (optimization) of all numbers; minimal for full coverage
GrayNo Resultsany calculation that has no results

Digits 1 2 3 4 5 6
Average Steps
Full Cycle		1.99 step
avg		27.51 step
avg		22.84 step
avg		13.26 step
avg		23.86 step
avg		355.31 step
avg
Step Frequency
Full Cycle
0s:
1s:
2s:
0×
1×
104×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
0×
1×
104×
52×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
13×
1,365×
1,326×
2,704×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
0×
1×
104×
5,617×
2,805×
5,608×
5,598×
5,584×
5,566×
5,544×
5,518×
5,488×
5,454×
5,416×
5,374×
5,328×
5,278×
5,224×
5,166×
5,104×
5,038×
4,968×
4,894×
4,816×
4,734×
4,648×
4,558×
4,464×
4,366×
4,264×
4,158×
4,048×
3,934×
3,816×
3,694×
3,568×
3,438×
3,304×
3,166×
3,024×
2,878×
2,728×
2,574×
2,416×
2,254×
2,088×
1,918×
1,744×
1,566×
1,384×
1,198×
1,008×
814×
616×
414×
208×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
0×
1×
1,910×
7,609×
23,636×
45,111×
45,192×
79,378×
206,538×
5,017×
0×
0×
0×
1,195,248×
3,600,188×
146,459×
2,808×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
0×
1×
45,464×
59,609×
104,570×
225,482×
376,740×
646,080×
825,972×
1,194,308×
1,130,706×
2,050,568×
1,971,129×
2,943,938×
2,338,505×
3,859,350×
2,770,680×
4,844,325×
3,647,408×
5,676,985×
4,488,629×
6,419,947×
4,740,624×
6,412,187×
4,957,013×
6,517,646×
4,699,304×
5,884,898×
4,357,879×
5,134,562×
3,871,894×
4,437,428×
3,309,627×
3,452,175×
2,774,265×
2,592,824×
2,074,457×
1,815,083×
1,380,478×
1,087,358×
789,074×
507,512×
272,332×
111,009×
28,246×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
56s:
57s:
58s:
59s:
60s:
61s:
62s:
63s:
64s:
65s:
66s:
67s:
68s:
69s:
70s:
71s:
72s:
73s:
74s:
75s:
76s:
77s:
78s:
79s:
80s:
81s:
82s:
83s:
84s:
85s:
86s:
87s:
88s:
89s:
90s:
91s:
92s:
93s:
94s:
95s:
96s:
97s:
98s:
99s:
100s:
101s:
102s:
103s:
104s:
105s:
106s:
107s:
108s:
109s:
110s:
111s:
112s:
113s:
114s:
115s:
116s:
117s:
118s:
119s:
120s:
121s:
122s:
123s:
124s:
125s:
126s:
127s:
128s:
129s:
130s:
131s:
132s:
133s:
134s:
135s:
136s:
137s:
138s:
139s:
140s:
141s:
142s:
143s:
144s:
145s:
146s:
147s:
148s:
149s:
150s:
151s:
152s:
153s:
154s:
155s:
156s:
157s:
158s:
159s:
160s:
161s:
162s:
163s:
164s:
165s:
166s:
167s:
168s:
169s:
170s:
171s:
172s:
173s:
174s:
175s:
176s:
177s:
178s:
179s:
180s:
181s:
182s:
183s:
184s:
185s:
186s:
187s:
188s:
189s:
190s:
191s:
192s:
193s:
194s:
195s:
196s:
197s:
198s:
199s:
200s:
201s:
202s:
203s:
204s:
205s:
206s:
207s:
208s:
209s:
210s:
211s:
212s:
213s:
214s:
215s:
216s:
217s:
218s:
219s:
220s:
221s:
222s:
223s:
224s:
225s:
226s:
227s:
228s:
229s:
230s:
231s:
232s:
233s:
234s:
235s:
236s:
237s:
238s:
239s:
240s:
241s:
242s:
243s:
244s:
245s:
246s:
247s:
248s:
249s:
250s:
251s:
252s:
253s:
254s:
255s:
256s:
257s:
258s:
259s:
260s:
261s:
262s:
263s:
264s:
265s:
266s:
267s:
268s:
269s:
270s:
271s:
272s:
273s:
274s:
275s:
276s:
277s:
278s:
279s:
280s:
281s:
282s:
283s:
284s:
285s:
286s:
287s:
288s:
289s:
290s:
291s:
292s:
293s:
294s:
295s:
296s:
297s:
298s:
299s:
300s:
301s:
302s:
303s:
304s:
305s:
306s:
307s:
308s:
309s:
310s:
311s:
312s:
313s:
314s:
315s:
316s:
317s:
318s:
319s:
320s:
321s:
322s:
323s:
324s:
325s:
326s:
327s:
328s:
329s:
330s:
331s:
332s:
333s:
334s:
335s:
336s:
337s:
338s:
339s:
340s:
341s:
342s:
343s:
344s:
345s:
346s:
347s:
348s:
349s:
350s:
351s:
352s:
353s:
354s:
355s:
356s:
357s:
358s:
359s:
360s:
361s:
362s:
363s:
364s:
365s:
366s:
367s:
368s:
369s:
370s:
371s:
372s:
373s:
374s:
375s:
376s:
377s:
378s:
379s:
380s:
381s:
382s:
383s:
384s:
385s:
386s:
387s:
388s:
389s:
390s:
391s:
392s:
393s:
394s:
395s:
396s:
397s:
398s:
399s:
400s:
401s:
402s:
403s:
404s:
405s:
406s:
407s:
408s:
409s:
410s:
411s:
412s:
413s:
414s:
415s:
416s:
417s:
418s:
419s:
420s:
421s:
422s:
423s:
424s:
425s:
426s:
427s:
428s:
429s:
430s:
431s:
432s:
433s:
434s:
435s:
436s:
437s:
438s:
439s:
440s:
441s:
442s:
443s:
444s:
445s:
446s:
447s:
448s:
449s:
450s:
451s:
452s:
453s:
454s:
455s:
456s:
457s:
458s:
459s:
460s:
461s:
462s:
463s:
464s:
465s:
466s:
467s:
468s:
469s:
470s:
471s:
472s:
473s:
474s:
475s:
476s:
477s:
478s:
479s:
480s:
481s:
482s:
483s:
484s:
485s:
486s:
487s:
488s:
489s:
490s:
491s:
492s:
493s:
494s:
495s:
496s:
497s:
498s:
499s:
500s:
501s:
502s:
503s:
504s:
505s:
506s:
507s:
508s:
509s:
510s:
511s:
512s:
513s:
514s:
515s:
516s:
517s:
518s:
519s:
520s:
521s:
522s:
0×
1×
104×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
171,560×
551,681×
965,578×
557,375×
754,842×
828,116×
902,966×
1,287,560×
2,211,510×
2,046,317×
2,599,186×
2,961,946×
2,448,469×
1,596,556×
1,490,143×
1,735,856×
1,906,032×
1,854,899×
1,292,056×
1,118,478×
1,392,715×
1,537,936×
1,624,401×
991,828×
975,390×
1,211,833×
2,363,343×
2,698,881×
3,934,786×
3,805,253×
4,485,736×
4,846,125×
4,226,856×
4,249,953×
4,995,028×
6,335,732×
5,565,743×
4,344,200×
4,025,908×
3,748,346×
3,386,630×
3,191,646×
2,790,937×
2,905,478×
3,001,147×
2,575,224×
3,378,553×
3,124,016×
4,174,323×
3,993,949×
4,302,311×
2,595,658×
1,649,051×
1,390,804×
1,540,318×
2,008,727×
2,444,388×
2,105,775×
2,438,994×
2,547,693×
2,021,618×
2,357,901×
3,037,752×
3,216,444×
3,128,847×
2,951,476×
3,877,856×
3,967,933×
3,446,667×
3,982,049×
3,126,450×
3,394,129×
3,520,919×
2,690,901×
2,558,250×
2,573,599×
3,239,328×
3,406,530×
3,600,263×
3,943,859×
2,955,544×
3,064,495×
2,678,513×
3,115,693×
2,971,536×
2,961,927×
3,010,415×
2,324,912×
2,344,539×
2,552,168×
2,422,750×
2,903,424×
2,727,849×
2,306,582×
1,271,925×
1,005,001×
1,168,211×
1,334,939×
573,608×
591,730×
563,039×
975,464×
755,898×
518,362×
459,452×
381,406×
365,262×
294,262×
287,198×
424,033×
109,095×
187,852×
315,702×
350,876×
476,587×
497,720×
486,823×
387,540×
555,273×
402,008×
229,089×
330,418×
428,003×
435,454×
710,608×
689,165×
379,193×
497,659×
300,099×
308,611×
551,129×
518,596×
690,518×
955,040×
771,878×
247,843×
159,012×
534,646×
699,963×
1,120,417×
889,815×
1,172,951×
462,187×
318,676×
448,058×
278,064×
481,870×
1,170,137×
1,104,359×
614,230×
542,607×
452,747×
359,150×
263,419×
242,708×
213,503×
381,789×
479,568×
345,087×
417,065×
646,067×
1,114,378×
1,068,310×
851,439×
946,025×
1,106,600×
269,120×
405,663×
382,513×
150,727×
382,414×
460,650×
279,123×
337,713×
250,526×
48,274×
45,102×
103,034×
1,080×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
3,799,759×
10,817,051×
20,758,852×
15,424,120×
18,464,743×
15,836,848×
15,809,821×
15,192,466×
18,408,826×
18,811,310×
18,772,514×
18,314,425×
16,988,213×
15,691,384×
16,155,675×
16,667,367×
16,817,023×
15,473,494×
16,352,912×
17,190,151×
16,293,327×
15,371,584×
16,805,281×
17,474,337×
16,642,413×
16,852,832×
17,424,361×
19,803,221×
18,039,240×
17,401,605×
15,103,796×
16,002,781×
16,968,734×
16,534,338×
16,972,689×
17,457,563×
16,522,232×
16,434,611×
17,210,282×
18,210,061×
16,706,174×
18,577,973×
17,077,469×
17,526,323×
17,783,287×
18,106,249×
17,017,558×
18,142,914×
18,152,500×
17,075,630×
17,098,126×
17,818,495×
18,760,998×
18,031,442×
18,310,710×
17,984,677×
17,509,525×
17,835,517×
17,210,559×
16,904,364×
16,672,982×
18,509,224×
19,872,780×
18,695,281×
18,269,335×
16,417,660×
16,553,991×
15,692,829×
16,905,327×
16,964,131×
17,966,782×
16,428,626×
18,660,875×
17,072,421×
17,990,152×
18,680,827×
19,225,755×
17,564,534×
17,223,938×
16,793,054×
16,294,383×
17,631,859×
16,225,469×
14,076,330×
11,646,821×
12,385,555×
11,503,393×
12,168,333×
13,218,350×
11,830,018×
11,585,407×
12,832,712×
13,097,719×
13,106,036×
11,933,281×
12,369,919×
11,912,218×
9,188,522×
10,752,367×
11,168,450×
10,430,672×
10,868,655×
8,545,941×
9,344,513×
8,273,326×
8,168,379×
7,991,512×
8,138,452×
7,204,478×
7,001,866×
5,876,087×
4,539,212×
3,638,192×
3,145,142×
3,619,509×
3,548,134×
3,508,384×
4,000,402×
3,814,492×
3,967,105×
4,230,285×
3,796,561×
4,039,317×
4,368,790×
4,890,529×
3,842,703×
3,361,518×
3,323,229×
2,104,108×
2,045,921×
2,126,375×
2,294,493×
2,106,305×
1,661,315×
1,753,910×
2,241,768×
2,674,089×
2,239,013×
1,981,265×
2,144,839×
2,178,632×
2,350,781×
2,001,274×
2,020,038×
2,092,362×
1,497,332×
1,367,024×
832,580×
1,392,790×
1,313,714×
740,294×
1,011,150×
1,288,489×
1,221,311×
967,016×
757,640×
725,997×
404,807×
584,914×
599,326×
582,450×
682,279×
529,848×
529,224×
660,664×
731,198×
831,008×
747,857×
550,764×
621,334×
790,672×
924,137×
825,183×
609,848×
782,140×
492,132×
544,755×
385,909×
457,671×
30,608×