Base 23: Iteration Frequencies: Full Cycle

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789ABCDEF
GHIJKLMNOPQRSTUV

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 9 10 11 12 13 14 15
Average Steps
Full Cycle
1.96 step
avg
4.61 step
avg
7.35 step
avg
12.91 step
avg
10.52 step
avg
19.86 step
avg
19.49 step
avg
37.30 step
avg
47.76 step
avg
52.76 step
avg
53.65 step
avg
46.88 step
avg
101.57 step
avg
43.23 step
avg
79.87 step
avg
Step Frequency
Full Cycle
0s:
1s:
2s:


22×
0s:
1s:
2s:
3s:
4s:
5s:
6s:


36×
18×
44×
88×
88×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:


22×
287×
140×
278×
268×
254×
236×
214×
188×
158×
124×
86×
44×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:


22×









3,300×
9,515×
1,969×
143×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:


22×


2,272×
4,122×
4,787×
9,388×
11,611×
11,914×
9,772×
6,562×
5,419×
4,489×
5,336×
3,901×
1,134×
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:


22×



791×
1,639×
2,399×
3,312×
5,603×
7,233×
15,404×
11,746×
15,721×
15,416×
16,734×
33,383×
39,978×
24,464×
21,189×
14,060×
19,018×
19,505×
30,892×
24,928×
11,131×
6,546×
10,449×
8,899×
6,462×
4,691×
3,316×
1,736×
72×
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:


22×






12,420×
26,907×
53,610×
55,945×
92,320×
110,037×
110,384×
147,256×
171,635×
154,478×
98,220×
85,315×
62,170×
33,330×
21,940×
6,441×
6,164×
13,196×
18,639×
25,135×
31,829×
46,637×
35,991×
19,757×
26,068×
11,298×
21,152×
30,079×
25,319×
6,362×
723×
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:


22×



791×
2,749×
6,883×
7,803×
1,001×
3,221×
4,736×
1,104×
3,591×
4,764×
1,810×


11,149×
39,968×
160,980×
104,969×
92,330×
170,840×
189,006×
275,623×
290,098×
255,045×
250,257×
203,203×
228,963×
273,499×
221,507×
197,011×
173,400×
109,081×
97,622×
71,160×
50,033×
54,967×
81,451×
70,357×
82,424×
128,449×
154,759×
237,288×
243,885×
158,537×
112,461×
83,449×
105,553×
157,577×
95,883×
128,747×
106,190×
71,948×
68,398×
29,425×
15,554×
11,194×
10,881×
7,475×
30,682×
43,494×
23,888×
21,714×
4,843×
2,300×
3,760×
1,172×
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:


4,022×
10,888×
30,552×
28,394×
22,821×
11,312×
17,340×
8,711×





























146,104×
480,867×
840,082×
1,293,293×
1,664,697×
1,741,304×
1,649,515×
1,469,649×
1,351,297×
1,217,686×
1,145,468×
962,680×
1,071,293×
1,026,714×
958,610×
510,768×
311,933×
380,910×
415,866×
365,573×
247,867×
290,306×
136,575×
110,938×
94,593×
58,338×
52,723×
10,788×
10,560×
5,348×
3,689×
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:


22×



791×
3,859×
14,688×
8,133×
2,353×
405×
3,322×








29,323×
203,383×
1,265,794×
1,200,841×
1,134,767×
784,294×
543,228×
440,238×
194,936×
156,406×
145,933×
93,533×
158,871×
377,614×
1,218,208×
1,247,068×
1,437,423×
806,711×
533,605×
348,579×
463,363×
519,289×
885,393×
2,957,636×
3,190,521×
3,374,628×
3,769,382×
3,786,713×
3,291,684×
2,837,627×
2,085,019×
1,614,334×
1,835,274×
1,503,164×
1,383,526×
1,112,499×
749,538×
759,997×
627,150×
483,456×
447,735×
437,094×
656,315×
1,004,635×
813,024×
622,787×
665,200×
733,016×
880,576×
691,979×
401,750×
489,865×
495,479×
448,704×
266,672×
146,784×
107,774×
65,503×
68,025×
93,743×
75,395×
25,931×
14,668×
14,680×
44,565×
20,436×
63,308×
72,216×
64,085×
226,668×
124,763×
147,465×
54,658×
18,898×
111,469×
268,389×
527,619×
462,375×
342,515×
221,835×
236,719×
272,218×
188,529×
207,068×
152,358×
68,787×
62,003×
70,437×
43,643×
37,076×
196,453×
240,767×
259,321×
180,557×
114,505×
106,343×
39,990×
10,499×
1,455×
392×
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:


22×



74,472×
423,061×
920,381×
1,363,816×
1,491,230×
1,321,083×
1,411,305×
1,078,397×
709,773×
868,641×
668,189×
414,521×
460,832×
540,868×
337,393×
324,255×
593,712×
978,228×
1,205,117×
2,177,237×
2,198,248×
2,476,066×
2,305,953×
2,509,166×
2,942,064×
2,199,515×
1,653,548×
1,543,950×
1,015,468×
870,377×
1,049,724×
767,758×
651,999×
225,775×
9,206×









389,319×
1,855,350×
3,752,260×
5,001,880×
6,388,657×
7,407,220×
7,298,934×
7,461,991×
8,644,399×
9,418,137×
11,937,236×
14,021,843×
13,583,720×
12,243,521×
9,378,434×
6,937,844×
5,404,459×
4,450,048×
3,856,951×
2,866,640×
1,947,581×
1,454,295×
1,003,018×
1,211,907×
1,275,846×
946,375×
560,868×
444,627×
492,276×
414,403×
601,078×
543,283×
429,949×
124,383×
6,507×
86×
44×
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:


5,422×
94,988×
1,376,702×
2,978,866×
2,720,612×
2,329,866×
1,266,544×
809,061×
678,114×
961,945×
866,854×
538,274×
361,103×
1,303,188×
7,056,380×
10,588,337×
10,873,332×
10,607,444×
11,901,995×
15,755,753×
15,297,345×
14,761,069×
14,458,956×
13,103,377×
12,154,005×
10,633,303×
7,693,405×
8,807,684×
8,841,874×
10,859,426×
12,679,025×
15,195,477×
15,551,026×
16,105,972×
16,973,425×
17,776,588×
15,505,895×
13,072,102×
8,898,748×
8,518,075×
7,410,169×
5,323,148×
5,845,001×
5,394,258×
4,742,657×
4,854,455×
2,864,181×
3,108,840×
2,152,759×
2,433,007×
2,910,456×
3,834,458×
3,025,043×
1,848,167×
1,124,963×
1,395,525×
853,306×
823,454×
585,790×
848,928×
1,671,178×
1,577,493×
2,944,818×
1,395,512×
1,081,821×
530,782×
951,334×
899,468×
813,834×
788,759×
1,300,858×
1,969,011×
3,103,269×
4,948,576×
7,712,734×
5,721,655×
5,877,255×
4,335,982×
4,152,660×
6,325,741×
5,233,937×
4,401,628×
3,643,114×
4,267,837×
4,910,327×
4,514,385×
3,567,593×
2,612,731×
1,535,814×
1,276,153×
530,439×
603,937×
389,164×
358,684×
750,377×
1,724,260×
1,551,769×
1,664,461×
2,002,513×
1,056,767×
1,790,145×
2,145,874×
1,105,671×
1,089,312×
1,497,717×
1,301,686×
1,707,337×
2,394,506×
2,664,610×
3,065,436×
3,457,231×
1,824,874×
1,641,844×
1,980,825×
1,414,133×
1,099,146×
1,290,782×
790,420×
1,310,753×
1,095,752×
990,477×
1,091,570×
1,271,738×
1,224,723×
855,772×
751,361×
809,533×
321,910×
152,513×
125,504×
45,604×
6,528×
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:


22×
13,720×
70,376×
15,792×














































































1,212,492×
10,011,154×
25,510,203×
35,147,843×
48,792,101×
50,965,558×
51,016,476×
57,884,580×
62,011,792×
64,849,631×
63,046,262×
57,788,793×
58,181,970×
62,679,218×
60,989,524×
56,169,342×
49,675,439×
47,356,874×
48,906,393×
41,041,953×
35,861,819×
33,314,594×
37,798,340×
33,703,596×
31,854,390×
33,555,142×
32,270,705×
30,078,182×
25,304,652×
20,232,873×
14,274,776×
9,113,086×
6,814,398×
8,044,194×
9,651,529×
11,420,905×
12,479,178×
14,788,708×
13,109,264×
14,532,488×
15,147,460×
11,665,135×
11,195,955×
10,160,881×
9,565,436×
7,209,767×
5,229,616×
3,618,069×
3,182,228×
2,620,693×
1,964,915×
1,568,326×
2,179,345×
2,175,426×
2,231,109×
1,698,211×
1,240,340×
906,146×
693,866×
417,428×
123,288×
3,832×
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:


12,022×
96,996×
471,020×
1,820,644×
1,433,619×
1,484,337×
23,282,914×
61,827,249×
64,015,409×
78,740,206×
72,945,693×
54,941,226×
54,356,806×
51,493,077×
69,290,949×
65,819,514×
58,695,812×
45,988,103×
58,212,471×
63,998,619×
55,698,755×
43,083,710×
36,917,886×
33,958,504×
30,921,432×
36,536,555×
34,438,056×
31,228,153×
39,014,333×
45,173,746×
43,998,027×
66,240,989×
73,213,148×
94,715,424×
105,992,001×
116,513,496×
103,001,733×
89,677,870×
70,214,080×
61,107,812×
61,551,875×
47,531,145×
38,052,728×
41,758,260×
38,518,744×
36,183,562×
37,603,364×
41,731,499×
45,596,373×
37,042,359×
47,991,917×
43,904,576×
49,671,587×
49,746,779×
47,958,530×
45,956,290×
47,136,484×
43,465,695×
39,789,121×
34,672,632×
35,387,199×
44,140,484×
45,556,757×
54,844,228×
54,821,632×
56,058,671×
42,676,364×
27,472,934×
31,709,196×
30,655,249×
27,149,788×
20,494,755×
15,700,473×
13,354,230×
10,298,762×
11,302,334×
15,504,781×
21,388,763×
25,140,192×
20,653,045×
22,905,001×
18,327,237×
24,915,849×
26,724,579×
17,948,875×
15,076,900×
14,243,400×
13,116,543×
12,598,168×
10,995,895×
6,433,447×
14,110,687×
14,189,487×
10,499,895×
7,525,879×
7,056,056×
11,341,444×
8,561,395×
5,716,223×
3,463,786×
3,125,452×
2,174,593×
1,475,560×
837,325×
535,296×
719,840×
569,798×
306,483×
51,021×
5,332×
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:


71,446×
1,037,016×
4,492,012×
6,844,309×
8,752,887×
15,741,474×
17,535,854×
19,097,262×
28,014,898×
34,224,393×
49,284,171×
74,582,275×
66,295,422×
60,635,675×
51,495,623×
39,245,688×
32,518,854×
28,750,924×
18,963,673×
25,523,716×
30,210,254×
42,194,441×
49,910,563×
66,500,252×
60,947,221×
58,490,388×
53,466,435×
61,886,127×
63,738,373×
59,242,169×
62,965,108×
51,148,825×
40,224,415×
51,627,297×
51,614,347×
48,643,423×
50,101,419×
46,970,400×
40,455,647×
45,320,061×
48,491,558×
44,881,482×
36,391,742×
22,978,345×
11,750,020×
7,931,104×
11,305,650×
12,271,744×
8,503,900×
7,878,211×
5,914,707×
2,728,830×
1,431,744×
1,465,708×
695,392×
957,661×
43,904×













1,811,956×
25,895,765×
75,262,499×
115,811,790×
166,074,267×
220,495,368×
247,282,859×
300,012,408×
304,668,109×
268,831,199×
248,465,843×
236,660,732×
213,067,696×
199,988,131×
190,792,270×
184,395,528×
205,728,102×
228,397,146×
241,967,502×
241,304,209×
246,124,817×
252,758,198×
234,073,652×
176,775,966×
150,559,095×
147,236,396×
170,103,444×
157,850,951×
142,629,200×
136,154,470×
147,296,806×
148,383,277×
164,199,451×
135,131,183×
107,030,181×
90,419,661×
81,289,543×
67,523,983×
63,527,892×
63,911,371×
61,684,604×
67,075,283×
60,213,676×
63,933,282×
47,707,052×
43,630,612×
39,073,860×
31,679,955×
30,543,133×
37,732,104×
30,352,423×
25,631,392×
22,322,636×
23,282,727×
32,323,581×
31,420,594×
22,661,009×
16,724,470×
12,670,293×
6,629,905×
7,620,741×
3,392,525×
1,323,277×
291,270×