Base 91: Iteration Frequencies: Full Cycle

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789ABCDEF
GHIJKLMNOPQRSTUV
WXYZabcdefghijkl
mnopqrstuvwxyzΑΒ
ΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣ
ΤΥΦΧΨΩαβγδεζηθικ

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 7
Average Steps
Full Cycle		1.99 step
avg		13.52 step
avg		20.21 step
avg		13.44 step
avg		22.02 step
avg		183.92 step
avg		29.35 step
avg
Step Frequency
Full Cycle
0s:
1s:
2s:
0×
1×
90×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
0×
1×
90×
45×
90×
0×
0×
0×
0×
0×
0×
6×
511×
473×
990×
1,980×
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:
0×
1×
90×
4,231×
2,112×
4,222×
4,212×
4,198×
4,180×
4,158×
4,132×
4,102×
4,068×
4,030×
3,988×
3,942×
3,892×
3,838×
3,780×
3,718×
3,652×
3,582×
3,508×
3,430×
3,348×
3,262×
3,172×
3,078×
2,980×
2,878×
2,772×
2,662×
2,548×
2,430×
2,308×
2,182×
2,052×
1,918×
1,780×
1,638×
1,492×
1,342×
1,188×
1,030×
868×
702×
532×
358×
180×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
0×
1×
90×
0×
6,326×
12,804×
0×
42,010×
106,036×
987×
0×
0×
0×
695,484×
2,083,250×
100,398×
2,115×
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:
0×
1×
90×
0×
89,007×
209,302×
175,567×
260,328×
469,605×
612,793×
901,403×
960,181×
1,375,734×
1,160,371×
1,897,600×
1,391,660×
2,545,436×
1,846,364×
3,171,663×
2,366,780×
3,741,540×
2,949,545×
3,934,615×
2,979,380×
3,768,582×
2,659,012×
3,426,798×
2,402,943×
2,929,455×
2,061,698×
2,180,342×
1,501,051×
1,426,368×
972,494×
701,289×
443,099×
257,050×
119,273×
42,560×
9,540×
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:
0×
1×
90×
0×
12,208×
29,750×
63,296×
110,306×
448,177×
1,566,800×
682,274×
294,288×
553,730×
373,427×
274,059×
167,762×
119,789×
125,156×
314,546×
125,043×
136,200×
76,445×
94,119×
29,520×
60,623×
36,468×
44,435×
103,932×
66,211×
38,056×
7,132×
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×
1,043,415×
3,055,845×
6,853,288×
17,726,513×
8,929,152×
9,524,877×
7,831,673×
9,969,745×
9,439,072×
9,704,133×
9,926,010×
11,260,715×
10,736,012×
11,223,754×
11,335,097×
10,647,206×
10,561,710×
10,396,896×
8,650,074×
8,450,946×
10,794,761×
10,837,177×
8,528,467×
7,615,655×
7,931,449×
7,554,680×
7,647,063×
8,562,745×
7,429,717×
6,875,169×
6,889,492×
7,373,210×
7,959,176×
7,471,877×
5,734,191×
5,249,061×
4,555,314×
4,372,065×
4,635,979×
3,868,713×
4,602,109×
6,116,612×
5,048,779×
4,962,961×
4,333,331×
5,083,603×
5,475,215×
6,172,663×
6,367,174×
6,582,765×
6,463,082×
6,652,250×
7,944,847×
6,873,030×
5,587,122×
4,678,456×
4,683,932×
5,290,795×
6,197,724×
6,783,649×
7,550,714×
7,565,638×
7,454,570×
6,851,122×
7,326,766×
5,637,134×
6,074,110×
5,695,057×
6,532,399×
5,982,247×
7,081,752×
6,660,558×
6,268,946×
6,089,497×
6,435,713×
7,915,097×
8,365,343×
8,886,412×
7,430,650×
7,058,353×
7,067,213×
5,817,539×
5,593,631×
7,002,193×
8,031,185×
7,689,370×
6,857,806×
5,855,658×
4,640,146×
5,967,384×
5,820,753×
5,653,540×
4,926,388×
7,341,422×
6,998,839×
5,730,123×
5,162,838×
4,199,176×
4,027,663×
5,111,698×
4,214,669×
4,726,247×
4,943,947×
5,016,540×
5,876,781×
5,671,065×
5,355,017×
6,176,019×
4,207,793×
3,454,774×
3,669,911×
3,498,755×
2,899,665×
3,722,828×
3,535,916×
3,139,537×
2,711,946×
2,517,204×
3,058,523×
4,000,754×
3,881,565×
4,060,827×
4,158,485×
3,837,648×
3,505,334×
3,747,942×
2,747,146×
3,395,072×
3,088,245×
2,836,472×
3,260,800×
2,802,131×
3,182,378×
2,461,050×
2,228,956×
2,238,718×
2,637,006×
2,111,658×
1,885,511×
1,818,436×
1,663,010×
1,827,364×
1,530,922×
1,219,174×
1,932,825×
1,806,220×
1,767,323×
1,645,088×
1,967,679×
2,196,857×
2,433,845×
1,783,355×
1,573,711×
1,187,242×
1,662,005×
2,108,703×
1,970,008×
2,312,895×
2,540,110×
2,802,121×
2,648,299×
3,535,244×
3,137,424×
2,297,396×
1,724,658×
1,372,352×
1,156,077×
1,242,875×
808,998×
908,182×
893,821×
596,626×
681,787×
535,544×
608,751×
846,547×
613,885×
869,507×
1,259,767×
1,337,290×
690,340×
228,494×
133,681×
46,287×
65,772×
238,324×
482,057×
174,990×
186,513×
658,571×
432,296×
350,190×
384,830×
303,936×
817,408×
463,082×
315,837×
312,597×
247,429×
515,799×
723,830×
814,632×
536,426×
515,414×
217,194×
132,827×
18,345×
90,497×
113,500×
106,095×
40,514×
39,160×
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:
0×
1×
90×
0×
0×
0×
0×
0×
0×
2,415,472×
5,770,396×
17,549,181×
27,350,737×
46,567,539×
67,104,114×
101,322,620×
142,921,993×
182,529,439×
226,515,897×
288,473,402×
338,014,262×
409,711,536×
477,295,660×
579,215,112×
632,215,933×
643,217,029×
687,319,558×
743,769,907×
770,869,288×
707,050,051×
702,960,367×
620,162,654×
588,892,148×
521,825,360×
433,152,863×
375,040,863×
313,312,270×
277,773,882×
230,505,333×
227,498,299×
194,341,207×
174,456,238×
155,353,810×
152,708,869×
158,847,173×
135,051,170×
109,703,825×
117,555,027×
86,894,588×
66,161,868×
46,727,085×
32,311,102×
15,051,455×
9,984,968×
2,392,092×
336,244×
40,807×