Base 15: Iteration Frequencies: Cycle Start

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789ABCDE

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 16 17 18 19 20 21 22 23 24
Average Steps
Cycle Start
0.93 step
avg
3.85 step
avg
3.79 step
avg
2.03 step
avg
7.20 step
avg
6.26 step
avg
6.11 step
avg
7.07 step
avg
20.32 step
avg
15.69 step
avg
12.89 step
avg
12.19 step
avg
35.10 step
avg
19.55 step
avg
12.99 step
avg
31.03 step
avg
18.43 step
avg
17.71 step
avg
12.49 step
avg
25.28 step
avg
14.13 step
avg
21.65 step
avg
12.36 step
avg
29.75 step
avg
Step Frequency
Cycle Start
0s:
1s:

14×
0s:
1s:
2s:
3s:
4s:
5s:

14×

14×
28×
56×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:

141×
60×
118×
108×
94×
76×
54×
28×
0s:
1s:
2s:
3s:
4s:

630×
1,757×
609×
63×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:

194×
419×
773×
1,482×
1,522×
964×
688×
622×
1,132×
1,942×
1,204×
498×
187×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:

991×
2,074×
3,528×
6,600×
6,284×
4,575×
2,259×
4,040×
1,264×
1,091×
2,200×
3,097×
687×
69×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:

5,693×
7,573×
11,146×
11,954×
14,028×
11,248×
13,969×
13,240×
13,389×
9,210×
3,257×
1,544×
28×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:

5,554×
18,291×
54,049×
40,442×
27,291×
24,656×
25,302×
24,639×
15,223×
13,690×
12,630×
12,139×
11,296×
8,563×
10,801×
11,112×
3,845×
246×
0s:
1s:
2s:
3s:
4s:
5s:
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34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:

8,850×
20,651×
36,393×
27,173×
18,348×
20,336×
15,046×
14,146×
16,107×
6,681×
8,783×
14,463×
23,025×
27,017×
25,837×
22,018×
22,977×
15,064×
21,423×
21,059×
29,674×
24,524×
19,286×
11,580×
12,347×
17,387×
37,077×
45,396×
42,462×
27,207×
27,141×
28,989×
36,312×
24,898×
16,484×
7,567×
9,228×
8,014×
4,915×
1,144×
160×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
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25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:

2,538×
16,133×
87,768×
79,803×
65,825×
49,863×
51,870×
63,844×
80,187×
75,014×
67,074×
56,843×
57,252×
87,402×
73,620×
99,815×
80,422×
77,684×
74,805×
77,108×
83,341×
108,829×
122,873×
87,772×
71,720×
56,613×
43,263×
22,356×
12,130×
6,857×
4,580×
3,145×
7,696×
2,189×
1,427×
1,594×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
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26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:

20,881×
91,131×
190,083×
239,648×
278,249×
306,985×
316,290×
268,781×
228,867×
226,034×
212,599×
180,637×
114,569×
143,511×
135,697×
120,432×
95,070×
94,667×
115,148×
147,283×
153,222×
111,778×
106,497×
95,172×
114,355×
85,655×
67,877×
53,039×
24,259×
17,957×
21,312×
25,244×
22,689×
15,936×
8,553×
6,472×
820×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
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22s:
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24s:
25s:
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28s:
29s:
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31s:
32s:
33s:

7,483×
79,477×
531,791×
598,152×
531,667×
482,187×
654,723×
699,361×
758,869×
635,620×
505,828×
355,173×
349,615×
335,660×
257,178×
192,647×
216,104×
216,072×
195,014×
360,441×
361,939×
361,478×
246,823×
197,938×
151,679×
98,693×
83,216×
71,913×
67,358×
31,167×
19,475×
2,622×
336×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
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60s:
61s:
62s:
63s:
64s:
65s:
66s:
67s:
68s:
69s:
70s:

7,010×
50,436×
108,216×
148,804×
327,040×
322,037×
320,561×
290,306×
306,010×
402,779×
256,467×
210,420×
294,205×
380,830×
475,611×
392,834×
503,954×
574,005×
461,834×
356,423×
269,353×
381,814×
275,290×
143,730×
136,052×
192,979×
224,906×
227,925×
289,287×
212,347×
178,398×
152,345×
104,435×
156,898×
288,057×
272,832×
290,522×
310,489×
410,286×
400,604×
431,988×
454,873×
410,068×
445,142×
496,841×
445,401×
504,572×
494,607×
448,908×
361,111×
220,251×
181,523×
161,645×
176,134×
215,638×
205,592×
202,419×
219,153×
245,125×
246,306×
342,419×
365,450×
348,984×
383,350×
359,602×
304,413×
202,142×
52,342×
21,133×
2,836×
0s:
1s:
2s:
3s:
4s:
5s:
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46s:
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48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
56s:
57s:
58s:
59s:
60s:
61s:

6,567×
84,207×
999,570×
1,519,908×
1,550,081×
1,467,048×
1,720,883×
1,635,486×
1,214,495×
1,107,191×
964,586×
1,003,097×
866,482×
922,406×
915,289×
1,480,188×
1,573,649×
1,479,330×
1,019,882×
772,877×
708,789×
1,089,622×
1,251,474×
1,284,626×
704,408×
1,044,311×
1,051,556×
1,266,372×
1,468,584×
1,480,441×
813,884×
567,143×
403,379×
423,306×
456,619×
445,818×
397,297×
200,051×
293,915×
451,128×
578,820×
383,082×
176,409×
141,137×
159,798×
88,166×
43,664×
36,249×
32,110×
39,020×
72,128×
52,940×
60,415×
43,368×
40,044×
27,324×
18,209×
8,691×
3,566×
5,354×
160×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
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23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:

84,182×
923,196×
2,511,861×
3,797,394×
4,082,493×
3,681,374×
4,879,510×
4,851,729×
4,518,097×
4,262,297×
4,330,395×
4,537,099×
3,951,110×
2,996,254×
2,397,311×
2,001,787×
2,093,420×
1,973,562×
2,386,859×
2,315,341×
2,820,952×
2,881,617×
2,272,168×
2,060,997×
1,410,256×
998,145×
857,951×
514,876×
445,832×
379,530×
191,543×
131,554×
17,923×
144×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
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8s:
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56s:
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62s:
63s:
64s:
65s:

16,866×
271,282×
2,135,369×
3,468,770×
3,961,733×
2,186,689×
3,244,030×
3,385,101×
2,941,701×
2,520,801×
2,542,205×
1,915,348×
2,148,286×
2,101,353×
1,666,706×
1,412,503×
1,699,775×
1,569,031×
1,344,950×
1,417,064×
915,418×
443,251×
1,305,448×
1,207,863×
3,209,951×
3,975,706×
4,257,158×
2,731,084×
2,066,890×
2,408,562×
4,001,389×
3,467,612×
3,705,432×
3,125,153×
2,511,819×
1,925,736×
2,361,048×
3,396,452×
4,107,694×
4,159,661×
3,641,273×
4,022,435×
3,575,283×
3,223,349×
1,932,052×
1,898,851×
1,751,344×
2,983,170×
4,596,699×
5,256,921×
3,340,488×
2,416,253×
1,707,622×
1,353,178×
1,574,138×
1,615,624×
1,508,318×
697,063×
360,531×
354,536×
264,723×
98,555×
11,784×
5,458×
136×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
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9s:
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34s:
35s:
36s:
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42s:
43s:
44s:

48,914×
926,992×
3,779,424×
7,754,654×
10,725,852×
12,115,614×
13,127,209×
12,152,640×
10,835,672×
9,380,332×
9,218,008×
8,393,314×
6,721,587×
4,874,255×
4,665,722×
3,908,686×
3,902,736×
4,292,935×
4,335,258×
7,159,787×
9,913,488×
11,975,608×
12,843,669×
12,914,923×
9,865,153×
8,409,783×
8,046,932×
6,995,292×
5,690,962×
5,263,111×
4,962,381×
5,738,712×
4,797,673×
4,329,972×
3,957,555×
3,636,343×
3,067,766×
1,871,564×
1,420,397×
789,453×
230,639×
72,603×
41,366×
27,588×
0s:
1s:
2s:
3s:
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38s:
39s:
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42s:
43s:
44s:
45s:

15,931×
356,504×
5,181,767×
10,078,090×
13,917,421×
15,784,992×
12,940,142×
11,412,151×
9,641,130×
16,461,560×
20,747,280×
29,454,735×
27,185,588×
29,739,525×
18,444,195×
16,188,068×
15,459,803×
18,865,068×
18,690,367×
20,367,023×
14,934,322×
8,908,022×
9,520,003×
8,210,298×
12,537,621×
15,652,867×
15,212,179×
11,682,197×
10,546,270×
9,049,202×
12,770,226×
8,244,546×
6,195,744×
4,120,728×
3,447,716×
2,688,982×
1,778,687×
1,781,207×
1,708,202×
834,520×
392,074×
128,654×
140,559×
19,013×
420×
0s:
1s:
2s:
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29s:
30s:

104,354×
2,948,668×
12,779,816×
27,944,075×
40,989,487×
45,565,810×
52,697,287×
46,700,712×
41,290,209×
42,393,545×
44,097,867×
47,716,223×
54,678,049×
49,220,772×
51,921,080×
48,303,587×
42,576,062×
41,128,470×
39,959,047×
33,812,992×
24,407,176×
15,050,792×
6,751,621×
3,243,246×
1,326,777×
738,233×
309,316×
89,170×
59,908×
4,848×
0s:
1s:
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69s:

56,847×
1,923,291×
26,497,477×
43,922,226×
54,814,396×
50,656,033×
33,098,243×
28,559,658×
20,866,869×
30,305,468×
30,534,123×
30,499,350×
32,384,581×
25,867,731×
17,142,845×
21,365,185×
28,717,692×
49,833,803×
45,283,028×
34,467,093×
24,741,190×
28,921,739×
24,790,557×
48,144,089×
56,730,553×
63,535,060×
34,313,164×
23,363,516×
15,926,869×
22,063,099×
25,421,358×
25,511,915×
15,420,894×
12,589,635×
8,892,823×
10,110,475×
10,419,535×
7,647,035×
6,110,088×
9,221,267×
14,109,469×
17,460,782×
14,414,846×
14,900,856×
16,600,033×
26,339,959×
28,137,481×
29,490,154×
28,082,724×
19,616,789×
14,718,855×
13,880,987×
16,144,467×
9,917,448×
9,732,038×
9,479,783×
9,459,197×
7,917,244×
5,607,929×
2,231,469×
1,237,219×
814,273×
430,688×
287,608×
191,851×
46,564×
19,328×
23,716×
11,082×
0s:
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69,882×
2,656,624×
17,914,628×
47,984,663×
69,013,491×
98,597,201×
127,725,130×
140,635,118×
153,673,240×
147,974,277×
133,281,410×
121,263,941×
115,218,444×
112,438,520×
116,087,855×
117,772,563×
102,352,377×
104,931,128×
97,339,968×
93,634,567×
83,871,894×
76,020,166×
59,322,990×
36,136,863×
26,446,568×
20,014,231×
16,733,273×
18,666,624×
16,679,631×
15,665,987×
11,796,385×
6,681,856×
5,882,604×
3,022,826×
1,415,641×
673,657×
328,312×
32,004×
2,860×
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56,301×
3,101,512×
52,284,891×
89,428,514×
106,014,343×
116,100,662×
87,986,353×
83,261,417×
86,726,305×
123,597,451×
145,103,808×
143,498,942×
129,024,060×
118,981,855×
98,389,566×
121,004,213×
153,761,709×
170,753,942×
118,226,421×
106,379,630×
88,231,700×
71,433,227×
68,247,059×
71,894,686×
52,611,891×
64,753,612×
54,992,939×
61,247,495×
59,361,535×
66,770,235×
66,818,950×
51,871,352×
58,400,880×
59,229,091×
106,213,726×
132,729,013×
160,404,723×
102,311,227×
76,805,611×
57,752,882×
75,225,120×
47,152,287×
26,730,049×
14,943,332×
11,438,603×
6,241,706×
4,837,732×
6,824,030×
8,689,659×
6,154,564×
1,906,364×
325,893×
62,447×
1,288×
396×
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67,862×
3,667,432×
26,119,400×
114,570,412×
179,553,629×
258,500,796×
316,249,976×
406,851,282×
490,880,796×
529,099,832×
504,705,686×
430,538,889×
411,795,369×
393,371,864×
388,284,697×
401,783,201×
377,818,200×
298,478,395×
245,490,002×
150,055,896×
72,883,909×
41,746,210×
22,914,374×
16,934,777×
10,480,580×
5,229,493×
4,601,061×
2,592,835×
1,007,872×
457,591×
196,166×
130,548×
18,479×
2,580×
2,208×
4,500×
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98,778×
8,179,113×
140,933,700×
255,621,912×
279,359,606×
273,031,877×
200,398,467×
186,870,821×
210,152,830×
269,268,893×
311,491,607×
251,369,593×
199,238,846×
191,739,108×
191,395,410×
208,693,586×
165,440,249×
152,536,797×
120,990,685×
95,324,414×
124,349,201×
166,957,002×
170,424,694×
171,073,019×
184,992,497×
142,657,452×
134,645,297×
116,348,523×
137,848,923×
176,847,617×
237,073,247×
140,152,996×
92,367,584×
76,845,431×
100,335,188×
81,325,229×
78,029,920×
67,417,773×
87,192,522×
104,964,203×
181,112,032×
150,843,343×
147,497,270×
135,609,603×
131,638,132×
147,545,044×
172,749,019×
206,657,017×
204,895,139×
262,788,741×
195,243,993×
148,194,935×
143,471,044×
181,852,662×
179,065,395×
129,715,501×
129,534,331×
88,194,818×
51,366,780×
62,191,087×
44,170,393×
38,733,269×
33,871,261×
28,461,777×
20,219,826×
31,983,164×
28,108,421×
15,359,127×
14,238,613×
16,459,096×
15,266,843×
13,142,486×
5,985,492×
3,259,663×
681,598×
717,640×
2,115,050×
1,870,960×
500,269×
154,337×
70,829×
27,001×
5,768×
720×