Base 61: Iteration Frequencies: Full Cycle

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789ABCDEF
GHIJKLMNOPQRSTUV
WXYZabcdefghijkl
mnopqrstuvwxyzΑΒ

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 8
Average Steps
Full Cycle		1.98 step
avg		7.00 step
avg		14.57 step
avg		31.31 step
avg		16.37 step
avg		85.52 step
avg		18.64 step
avg		192.66 step
avg
Step Frequency
Full Cycle
0s:
1s:
2s:
0×
1×
60×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
0×
1×
60×
30×
0×
8×
457×
435×
900×
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:
0×
1×
60×
1,921×
957×
1,912×
1,902×
1,888×
1,870×
1,848×
1,822×
1,792×
1,758×
1,720×
1,678×
1,632×
1,582×
1,528×
1,470×
1,408×
1,342×
1,272×
1,198×
1,120×
1,038×
952×
862×
768×
670×
568×
462×
352×
238×
120×
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:
0×
1×
60×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
5,270×
15,535×
361×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
147,560×
435,360×
30,269×
960×
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:
0×
1×
60×
0×
27,537×
68,742×
61,437×
86,168×
157,095×
202,223×
308,886×
313,191×
480,308×
449,306×
647,723×
543,668×
749,123×
640,672×
781,540×
577,699×
661,698×
381,697×
412,163×
276,165×
203,389×
126,559×
68,494×
27,849×
6,495×
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:
0×
1×
60×
5,868×
14,160×
16,750×
118,970×
369,461×
1,173,044×
662,730×
364,372×
265,186×
385,332×
576,321×
510,712×
542,127×
666,484×
732,997×
346,112×
417,356×
401,104×
279,254×
347,284×
315,839×
281,392×
466,471×
648,011×
807,159×
673,155×
659,655×
468,049×
504,678×
688,404×
640,908×
604,666×
646,282×
613,031×
447,120×
544,160×
681,254×
568,591×
658,767×
666,925×
410,564×
309,007×
265,494×
215,590×
226,502×
265,762×
246,732×
413,226×
386,591×
401,738×
400,535×
334,018×
178,068×
156,844×
191,642×
166,810×
165,700×
366,122×
556,079×
911,094×
845,295×
743,679×
790,057×
865,285×
1,060,793×
1,051,827×
894,519×
830,910×
730,105×
860,751×
957,735×
871,385×
702,696×
504,048×
557,882×
494,599×
456,295×
501,138×
383,656×
493,211×
415,689×
643,005×
865,330×
738,968×
913,485×
808,320×
843,353×
949,553×
844,614×
703,171×
654,054×
763,516×
812,703×
746,543×
811,782×
912,628×
724,967×
680,286×
774,450×
746,410×
775,898×
742,959×
929,105×
933,301×
906,256×
827,611×
822,042×
825,142×
689,302×
830,036×
897,145×
919,722×
939,626×
871,416×
1,033,263×
958,022×
722,519×
678,339×
728,706×
764,713×
592,394×
433,155×
291,765×
452,140×
585,142×
409,674×
397,465×
465,912×
578,740×
520,509×
352,375×
326,989×
330,985×
358,450×
307,605×
278,379×
221,794×
238,621×
148,389×
155,693×
280,801×
358,959×
407,144×
484,825×
404,252×
357,188×
261,057×
306,739×
444,329×
547,662×
409,282×
429,471×
561,661×
647,868×
617,298×
619,262×
392,996×
365,705×
372,432×
295,969×
349,278×
287,350×
245,748×
231,651×
217,105×
163,113×
95,539×
61,281×
46,411×
57,650×
430×
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:
0×
1×
60×
258,600×
662,428×
1,923,516×
4,823,128×
8,906,636×
13,011,212×
16,607,068×
23,087,900×
29,776,822×
38,298,492×
45,147,731×
51,631,006×
56,697,662×
59,649,914×
58,350,430×
60,811,278×
51,927,558×
44,062,543×
38,032,470×
36,283,940×
31,013,586×
31,526,260×
27,247,172×
27,193,953×
24,358,730×
22,238,549×
21,224,185×
16,159,266×
10,724,982×
7,013,574×
5,010,809×
2,610,805×
2,336,302×
589,480×
424,251×
25,909×
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:
0×
1×
60×
14,100×
56,880×
81,244×
604,920×
2,626,952×
14,747,487×
13,861,227×
6,005,540×
4,720,996×
3,291,936×
2,571,452×
5,586,529×
6,925,877×
8,593,673×
7,030,029×
8,358,497×
9,359,876×
15,167,030×
18,572,534×
12,200,661×
9,097,706×
8,039,108×
9,527,415×
11,494,846×
9,466,528×
5,885,768×
4,330,036×
4,630,981×
1,391,819×
144,887×
214,875×
139,181×
95,383×
33,870×
110,214×
3,980×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
745,625×
3,259,221×
11,242,318×
17,076,860×
17,318,002×
16,123,694×
15,643,653×
16,793,180×
18,032,074×
12,052,535×
9,693,991×
3,683,834×
3,918,316×
3,530,317×
2,696,064×
1,372,736×
1,446,896×
1,634,960×
5,582,822×
21,738,268×
31,535,076×
34,617,177×
27,530,285×
27,629,019×
33,755,542×
30,161,972×
27,931,306×
22,747,249×
25,675,558×
21,154,944×
16,471,589×
14,345,203×
14,766,806×
11,812,791×
7,668,635×
6,778,213×
8,404,915×
8,359,500×
6,883,555×
6,077,887×
4,113,729×
7,076,177×
7,486,298×
5,061,949×
3,238,085×
5,262,817×
6,401,832×
10,577,474×
7,161,359×
5,898,483×
5,876,122×
7,217,006×
4,538,035×
2,934,920×
8,239,853×
30,228,535×
41,148,772×
41,602,090×
42,698,682×
50,074,265×
56,663,214×
47,703,633×
43,929,445×
38,006,435×
38,009,593×
32,208,787×
33,776,246×
30,915,403×
35,722,616×
34,696,747×
28,558,437×
29,542,131×
32,929,810×
34,091,784×
32,352,507×
24,617,267×
29,283,769×
25,132,314×
24,589,918×
17,371,348×
15,610,758×
20,165,088×
17,566,443×
19,336,720×
22,492,296×
21,570,906×
21,671,552×
19,242,517×
15,297,102×
17,943,343×
23,387,978×
25,381,204×
26,879,412×
30,837,813×
19,016,773×
16,559,100×
16,441,689×
25,518,920×
26,700,608×
31,081,438×
28,831,319×
32,598,067×
26,786,414×
25,524,759×
27,011,541×
32,156,302×
32,091,068×
29,485,961×
33,000,173×
29,957,324×
24,632,106×
25,970,949×
31,120,375×
36,694,492×
40,542,975×
30,511,782×
23,217,395×
29,103,255×
28,929,256×
29,211,103×
31,815,067×
31,450,807×
34,270,822×
38,350,044×
31,310,243×
31,049,801×
30,202,573×
29,703,768×
29,889,817×
29,963,451×
34,632,582×
29,210,966×
30,263,988×
39,559,932×
41,562,523×
42,288,550×
51,779,237×
52,107,035×
46,050,400×
48,710,715×
46,601,109×
51,968,616×
58,138,866×
53,340,956×
55,143,952×
44,396,570×
36,876,384×
43,176,798×
46,348,987×
52,882,638×
42,354,428×
45,014,144×
45,445,274×
47,298,692×
52,523,824×
56,259,344×
50,397,933×
45,786,530×
42,734,430×
47,543,582×
38,140,273×
36,976,346×
31,950,629×
34,326,288×
44,524,600×
40,741,593×
56,464,138×
61,616,778×
49,140,269×
54,411,024×
52,659,171×
51,063,008×
47,973,519×
44,593,816×
43,898,110×
31,025,095×
27,723,648×
27,337,624×
32,538,365×
30,928,698×
25,650,956×
33,374,340×
33,638,945×
28,126,968×
28,635,689×
30,471,457×
35,547,045×
51,896,123×
64,277,820×
60,583,929×
51,967,124×
41,642,665×
52,813,645×
47,401,364×
42,821,012×
57,463,476×
60,060,427×
46,932,937×
46,086,761×
41,029,267×
41,889,402×
50,942,397×
46,004,626×
32,976,230×
35,547,859×
34,731,637×
38,368,264×
35,412,964×
28,773,194×
31,060,349×
32,523,271×
40,928,400×
32,531,945×
26,187,569×
27,583,653×
30,556,275×
31,440,297×
34,852,729×
23,589,396×
34,510,591×
35,853,060×
25,140,303×
22,444,049×
14,811,822×
19,128,682×
20,998,734×
16,292,651×
9,586,170×
8,547,220×
9,734,168×
11,233,712×
6,661,209×
7,048,476×
7,423,170×
7,076,070×
4,856,603×
4,098,202×
3,662,703×
4,661,628×
4,611,336×
4,017,926×
3,860,677×
4,171,050×
2,967,608×
6,574,148×
8,908,491×
13,432,842×
6,306,312×
10,279,735×
6,924,714×
9,872,220×
9,294,594×
10,698,533×
14,444,162×
11,719,095×
12,147,627×
10,801,412×
12,020,120×
5,398,320×
4,137,978×
3,481,414×
3,567,317×
6,550,392×
6,019,422×
5,278,619×
5,165,961×
6,751,901×
9,783,657×
7,243,135×
5,465,999×
4,108,189×
4,539,449×
6,905,079×
6,917,613×
4,966,774×
2,338,790×
3,426,727×
2,909,177×
804,406×
275,268×
627,443×
1,440,261×
651,628×
644,507×
617,152×
2,300,545×
212,034×
5,160×