Base 53: 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
8.85 step
avg
13.06 step
avg
27.29 step
avg
15.55 step
avg
79.40 step
avg
17.44 step
avg
79.67 step
avg
Step Frequency
Full Cycle
0s:
1s:
2s:


52×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:


86×
44×
132×
75×
156×



230×
233×
468×
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:


52×
1,457×
725×
1,448×
1,438×
1,424×
1,406×
1,384×
1,358×
1,328×
1,294×
1,256×
1,214×
1,168×
1,118×
1,064×
1,006×
944×
878×
808×
734×
656×
574×
488×
398×
304×
206×
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:
30s:


52×











3,378×
10,953×
413×










84,372×
247,500×
19,893×
728×
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:


52×


24,552×
32,782×
39,512×
90,452×
122,928×
195,802×
220,444×
306,420×
331,555×
416,479×
358,616×
394,021×
320,493×
291,318×
263,218×
215,411×
185,388×
124,377×
87,155×
60,241×
55,995×
37,454×
12,440×
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:


3,796×
7,956×
42,298×
60,867×
137,412×
113,038×
108,609×
150,380×
176,621×
170,564×
193,721×
205,762×
201,028×
245,176×
271,322×
340,949×
309,962×
254,327×
241,632×
386,656×
434,881×
440,213×
408,846×
367,031×
481,079×
560,898×
599,535×
559,781×
450,878×
424,923×
546,347×
548,759×
520,067×
509,406×
534,396×
360,955×
321,973×
397,471×
457,988×
429,940×
491,800×
337,722×
233,612×
314,912×
252,931×
279,479×
264,392×
211,543×
276,072×
283,632×
241,772×
220,692×
245,469×
228,910×
167,148×
126,325×
151,164×
69,446×
88,399×
85,961×
110,202×
116,187×
160,805×
109,994×
181,731×
150,036×
147,378×
78,916×
93,016×
99,991×
139,846×
137,864×
140,681×
145,053×
142,962×
196,959×
206,710×
247,484×
221,701×
207,409×
264,253×
276,576×
216,802×
310,526×
307,831×
271,976×
253,112×
240,695×
268,828×
276,563×
197,732×
203,036×
181,464×
189,459×
158,348×
183,081×
232,391×
277,074×
292,902×
389,529×
409,261×
376,070×
391,328×
399,391×
471,959×
470,017×
466,460×
375,421×
332,006×
325,424×
440,639×
354,284×
313,838×
267,787×
255,914×
240,244×
203,338×
261,930×
337,687×
314,930×
294,780×
336,868×
391,186×
393,939×
335,925×
298,363×
257,540×
157,683×
75,375×
70,737×
72,152×
58,170×
57,764×
37,658×
42,667×
51,073×
38,094×
55,438×
62,223×
27,149×
37,303×
52,070×
75,580×
90,403×
101,156×
134,292×
132,404×
124,771×
143,222×
144,245×
135,843×
122,948×
144,800×
138,751×
51,887×
34,158×
40,797×
73,398×
138,122×
129,813×
100,078×
44,947×
46,683×
53,075×
82,660×
128,067×
155,632×
156,113×
97,120×
116,604×
126,074×
152,253×
169,443×
49,550×
70,499×
101,924×
110,784×
114,075×
61,054×
40,366×
56,328×
22,350×
41,921×
31,834×
58,244×
63,733×
54,024×
33,815×
37,066×
30,141×
14,479×
33,670×
38,204×
13,989×
17,171×
24,129×
25,312×
27,073×
44,380×
22,080×
7,077×
12,870×
28,395×
14,994×
475×
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:


52×
150,956×
388,392×
1,083,240×
2,647,992×
4,883,822×
6,557,481×
9,116,112×
11,648,446×
15,190,682×
18,378,302×
21,407,227×
25,237,217×
27,342,493×
26,055,945×
23,881,739×
20,040,716×
14,875,331×
11,953,731×
11,032,757×
11,373,345×
12,434,989×
13,727,752×
12,794,904×
12,122,294×
8,089,054×
6,154,503×
3,923,860×
3,050,496×
2,623,671×
1,523,572×
848,519×
326,641×
203,975×
75,184×
4,053×
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:


52×




41,136×
252,103×
1,055,705×
2,643,139×
2,831,112×
4,234,842×
4,122,471×
3,794,755×
3,604,772×
2,747,830×
3,456,683×
4,797,609×
3,277,943×
2,798,772×
1,698,540×
2,388,915×
2,684,629×
4,073,041×
3,731,167×
1,733,392×
2,088,465×
5,134,624×
6,251,683×
7,115,723×
12,429,667×
12,210,419×
10,358,206×
14,600,499×
15,290,888×
18,278,237×
24,110,826×
23,472,532×
25,599,809×
26,130,251×
24,129,897×
19,075,025×
18,643,678×
17,320,787×
18,084,035×
21,723,084×
19,014,755×
17,775,413×
21,048,798×
30,763,372×
40,672,038×
42,507,668×
42,421,304×
32,717,759×
28,556,024×
29,651,017×
31,429,867×
31,609,516×
28,093,831×
23,223,413×
21,578,263×
21,748,319×
20,387,421×
16,026,134×
17,262,394×
16,402,618×
21,052,761×
24,871,230×
24,056,840×
23,658,509×
27,941,724×
33,983,795×
30,934,191×
30,676,762×
36,758,801×
31,494,723×
28,566,893×
25,139,765×
27,519,914×
23,028,008×
26,085,197×
31,187,526×
29,503,875×
34,829,030×
34,400,078×
40,957,646×
39,230,600×
40,885,925×
38,014,931×
29,365,397×
30,389,080×
32,563,932×
32,535,375×
26,699,282×
22,759,443×
23,355,321×
28,222,428×
28,221,665×
23,094,925×
27,540,001×
26,055,282×
23,349,793×
26,206,562×
33,994,062×
33,563,778×
23,338,666×
24,919,615×
19,043,168×
17,730,257×
19,790,412×
17,446,863×
21,241,628×
18,764,087×
15,369,822×
22,480,184×
24,337,128×
18,190,793×
20,213,227×
18,553,446×
13,199,091×
9,750,361×
10,999,216×
10,844,722×
8,812,019×
8,474,478×
9,152,473×
6,713,842×
6,663,083×
4,918,248×
4,043,286×
3,194,661×
3,554,892×
2,384,369×
1,640,495×
2,508,812×
4,356,775×
2,226,763×
2,939,238×
1,436,569×
1,430,983×
575,427×
802,311×
3,076,730×
2,419,083×
4,864,921×
3,241,936×
3,582,825×
2,916,514×
4,030,157×
4,490,970×
4,137,987×
5,866,011×
2,771,456×
2,413,041×
1,729,463×
1,312,764×
3,264,942×
4,192,131×
4,059,173×
5,002,710×
3,858,912×
2,195,064×
1,295,600×
714,378×
1,845,902×
2,335,043×
699,283×
748,522×
1,100,895×
1,723,690×
2,076,500×
706,705×
620,042×
492,497×
935,336×
518,441×
761,190×
3,241,510×
1,431,441×
550,194×
613,304×
706,560×
156,288×
140,968×
18,976×
80,736×
20,430×
13,380×
2,916×