Base 111: Iteration Frequencies: Cycle Start

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
Cycle Start		0.99 step
avg		4.85 step
avg		21.97 step
avg		1.78 step
avg		23.93 step
avg		71.17 step
avg
Step Frequency
Cycle Start
0s:
1s:
1×
110×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
3×
297×
196×
440×
880×
1,760×
2,640×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
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35s:
36s:
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38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
56s:
1×
6,381×
3,132×
6,262×
6,252×
6,238×
6,220×
6,198×
6,172×
6,142×
6,108×
6,070×
6,028×
5,982×
5,932×
5,878×
5,820×
5,758×
5,692×
5,622×
5,548×
5,470×
5,388×
5,302×
5,212×
5,118×
5,020×
4,918×
4,812×
4,702×
4,588×
4,470×
4,348×
4,222×
4,092×
3,958×
3,820×
3,678×
3,532×
3,382×
3,228×
3,070×
2,908×
2,742×
2,572×
2,398×
2,220×
2,038×
1,852×
1,662×
1,468×
1,270×
1,068×
862×
652×
438×
220×
0s:
1s:
2s:
3s:
4s:
1×
1,635,590×
4,852,485×
181,665×
3,135×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
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17s:
18s:
19s:
20s:
21s:
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23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
1×
53,538×
69,379×
122,386×
263,226×
440,596×
753,560×
965,664×
1,391,312×
1,322,134×
2,390,324×
2,304,923×
3,433,486×
2,738,519×
4,507,330×
3,166,761×
5,627,004×
3,939,244×
6,827,710×
5,228,381×
7,870,212×
5,802,062×
7,941,994×
6,088,779×
7,995,741×
6,124,579×
7,987,922×
5,699,416×
7,043,396×
5,288,156×
6,209,158×
4,565,812×
5,276,411×
3,982,576×
4,089,221×
3,331,596×
3,076,003×
2,454,537×
2,145,731×
1,636,437×
1,293,721×
936,638×
597,477×
324,862×
133,229×
34,784×
220×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
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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:
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156s:
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174s:
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205s:
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207s:
208s:
209s:
210s:
211s:
212s:
213s:
214s:
215s:
216s:
217s:
218s:
219s:
220s:
221s:
222s:
223s:
1×
1,836,069×
4,843,914×
9,556,611×
21,270,601×
38,814,349×
65,951,956×
29,248,136×
31,455,380×
34,466,688×
29,423,322×
28,328,487×
25,900,395×
23,918,494×
27,277,681×
23,865,670×
25,248,685×
23,945,193×
20,024,389×
23,111,945×
37,932,171×
38,383,993×
27,197,404×
24,980,310×
26,439,489×
25,643,396×
27,100,789×
26,399,381×
23,016,556×
18,682,442×
16,816,205×
18,065,073×
17,340,241×
16,145,646×
17,344,897×
17,782,572×
16,249,323×
18,908,141×
18,868,059×
27,245,448×
24,073,163×
16,366,164×
17,810,881×
19,503,215×
13,946,561×
15,120,261×
14,776,492×
15,500,598×
19,138,785×
27,812,831×
27,047,246×
20,857,286×
15,286,601×
16,590,759×
19,239,682×
17,009,249×
19,848,977×
20,444,138×
19,093,595×
22,416,225×
31,745,411×
23,599,090×
21,600,267×
24,050,123×
25,829,924×
24,053,027×
30,408,617×
23,019,600×
23,655,611×
17,856,262×
20,384,938×
26,357,985×
25,339,486×
18,663,998×
16,565,074×
16,001,082×
17,926,044×
21,805,104×
19,566,238×
21,786,540×
17,904,295×
21,232,048×
18,932,647×
20,016,302×
20,221,715×
16,449,503×
17,548,190×
15,241,039×
14,453,295×
14,555,866×
15,719,188×
17,619,410×
12,705,235×
12,443,291×
12,080,965×
13,998,108×
17,902,946×
22,393,256×
15,685,015×
14,181,388×
13,763,033×
19,832,940×
23,760,131×
16,975,061×
16,002,569×
13,917,894×
18,010,005×
29,043,811×
24,250,357×
19,935,246×
18,997,505×
18,036,143×
18,768,855×
15,620,058×
13,060,615×
16,838,867×
18,014,117×
19,916,593×
13,840,693×
10,999,607×
13,064,049×
12,890,767×
11,777,943×
11,231,309×
16,371,058×
14,012,439×
12,666,466×
9,547,677×
9,057,131×
7,467,860×
9,216,126×
8,668,321×
6,589,838×
5,343,843×
5,860,905×
5,889,207×
6,311,998×
6,228,057×
5,882,519×
6,093,210×
6,858,691×
6,154,064×
5,720,855×
6,829,492×
5,979,366×
5,774,261×
5,247,791×
4,948,515×
4,100,538×
4,843,435×
5,124,755×
5,208,214×
4,640,691×
4,625,001×
7,867,614×
14,715,279×
13,379,533×
11,195,100×
9,461,931×
13,969,009×
12,914,969×
9,968,130×
7,666,443×
7,855,319×
11,022,893×
17,535,066×
9,602,970×
6,604,995×
6,214,282×
5,078,106×
3,648,266×
3,168,153×
3,472,660×
3,385,680×
4,047,155×
2,528,568×
3,047,314×
2,010,505×
4,039,298×
2,010,187×
883,402×
1,206,982×
944,993×
1,165,131×
766,611×
826,844×
836,018×
730,186×
803,207×
639,228×
827,070×
837,783×
2,004,526×
949,157×
548,039×
645,358×
1,108,358×
1,153,955×
597,178×
287,439×
358,537×
1,543,553×
888,232×
906,421×
2,593,263×
1,631,654×
753,368×
833,000×
941,726×
612,576×
774,743×
1,248,653×
1,975,192×
2,122,817×
1,105,896×
671,528×
735,024×
284,457×
473,872×
131,302×
110,542×
22,434×
47,901×
47,121×