Base 23: Iteration Frequencies: Cycle Start

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
Cycle Start
0.96 step
avg
3.61 step
avg
5.36 step
avg
1.93 step
avg
6.53 step
avg
8.16 step
avg
11.49 step
avg
17.34 step
avg
10.01 step
avg
16.13 step
avg
13.52 step
avg
34.26 step
avg
18.57 step
avg
26.75 step
avg
21.92 step
avg
Step Frequency
Cycle Start
0s:
1s:

22×
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2s:
3s:
4s:
5s:

36×
18×
44×
88×
88×
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1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
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12s:

309×
140×
278×
268×
254×
236×
214×
188×
158×
124×
86×
44×
0s:
1s:
2s:
3s:
4s:

3,322×
9,515×
1,969×
143×
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1s:
2s:
3s:
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5s:
6s:
7s:
8s:
9s:
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13s:

2,294×
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:
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21s:

5,670×
12,093×
22,721×
43,128×
47,988×
34,044×
30,495×
18,112×
22,218×
22,120×
33,402×
26,189×
16,388×
6,546×
10,449×
8,899×
6,462×
4,691×
3,316×
1,736×
72×
0s:
1s:
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31s:

12,442×
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:
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28,646×
102,289×
328,346×
275,792×
226,823×
312,764×
274,619×
285,897×
365,161×
313,854×
265,041×
189,736×
131,349×
106,886×
86,137×
51,957×
71,502×
76,888×
63,718×
48,013×
55,222×
47,261×
77,409×
69,432×
66,509×
112,952×
154,967×
224,620×
259,115×
172,732×
125,733×
85,267×
98,870×
123,963×
71,847×
102,866×
86,786×
55,495×
53,732×
26,161×
15,164×
11,194×
10,881×
7,475×
30,682×
43,494×
23,888×
21,714×
4,843×
2,300×
3,760×
1,172×
0s:
1s:
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31s:

150,126×
491,755×
870,634×
1,321,687×
1,687,518×
1,752,616×
1,666,855×
1,478,360×
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:
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79s:

127,532×
850,421×
4,815,218×
4,790,135×
5,016,015×
4,262,861×
3,922,630×
3,472,075×
2,872,139×
2,292,245×
1,749,554×
1,890,438×
1,945,358×
1,956,478×
1,987,620×
1,633,463×
1,480,576×
1,298,410×
873,610×
794,038×
982,085×
1,015,817×
1,351,625×
988,269×
765,949×
755,922×
828,431×
963,953×
786,556×
440,913×
518,161×
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:
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532,973×
2,454,526×
4,927,537×
6,415,155×
7,855,714×
8,967,525×
8,514,794×
8,376,851×
9,542,709×
10,259,910×
12,363,452×
14,453,273×
14,149,947×
12,649,671×
9,724,780×
7,533,879×
6,382,687×
5,655,165×
6,034,188×
5,064,888×
4,423,647×
3,760,248×
3,512,184×
4,153,971×
3,475,361×
2,599,923×
2,104,818×
1,460,095×
1,362,653×
1,464,127×
1,368,836×
1,195,282×
655,724×
133,589×
6,507×
86×
44×
0s:
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109s:
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111s:
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114s:
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118s:
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120s:

59,398×
823,363×
7,875,835×
13,162,987×
13,281,131×
12,709,831×
12,936,776×
16,256,972×
15,867,012×
15,690,566×
15,317,378×
13,634,643×
12,461,923×
11,213,085×
8,275,505×
9,238,395×
9,165,548×
11,083,013×
12,890,335×
15,476,824×
15,647,821×
16,137,343×
16,977,457×
17,783,596×
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:
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1,226,234×
10,081,530×
25,525,995×
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:
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80s:
81s:

229,008×
4,048,765×
46,342,598×
99,902,651×
119,892,698×
141,361,605×
142,380,948×
124,435,084×
116,180,938×
103,701,204×
118,489,566×
113,198,433×
94,256,818×
74,900,769×
92,194,108×
95,424,351×
83,571,549×
72,363,023×
67,434,994×
64,941,466×
52,898,114×
57,541,224×
50,076,115×
45,565,701×
49,026,944×
56,076,235×
55,968,767×
60,600,182×
50,831,373×
58,400,409×
66,512,531×
71,138,330×
65,209,000×
62,707,877×
58,094,236×
52,879,816×
63,320,192×
49,016,229×
33,590,953×
36,041,434×
34,486,575×
31,816,632×
26,803,126×
26,019,722×
26,801,764×
24,962,285×
38,176,269×
43,384,143×
56,617,235×
64,860,568×
57,548,032×
54,341,912×
49,733,583×
54,066,194×
47,411,904×
29,678,914×
26,423,534×
26,682,759×
23,819,161×
27,354,113×
24,846,906×
13,344,882×
19,740,961×
17,211,530×
13,944,280×
10,787,565×
10,699,980×
13,357,124×
9,456,537×
6,882,100×
3,865,940×
3,238,765×
2,561,748×
1,581,127×
851,117×
535,811×
722,085×
570,055×
307,670×
51,021×
5,332×
0s:
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2,086,490×
30,321,905×
85,498,561×
130,083,815×
192,072,991×
259,466,395×
299,451,046×
374,175,519×
374,480,957×
331,891,889×
299,796,092×
280,722,771×
250,291,561×
235,893,236×
218,862,708×
215,222,468×
237,552,968×
271,102,507×
292,055,049×
307,807,533×
307,072,038×
311,248,586×
287,540,087×
238,662,093×
214,297,468×
206,478,565×
233,068,552×
208,999,776×
182,853,615×
187,781,767×
198,911,153×
197,026,700×
214,300,870×
182,101,583×
147,485,828×
135,739,722×
129,781,101×
112,405,465×
99,919,634×
86,889,716×
73,434,624×
75,006,387×
71,519,326×
76,205,026×
56,210,952×
51,508,823×
44,988,567×
34,408,785×
31,974,877×
39,197,812×
31,047,815×
26,589,053×
22,366,540×
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×