Base 26: 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
Average Steps
Full Cycle		1.96 step
avg		7.98 step
avg		7.44 step
avg		8.80 step
avg		11.04 step
avg		41.94 step
avg		16.01 step
avg		62.43 step
avg		22.70 step
avg		146.31 step
avg		25.16 step
avg		193.13 step
avg		26.07 step
avg
Step Frequency
Full Cycle
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0×
1×
25×
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0×
2×
41×
9×
38×
36×
0×
0×
0×
5×
108×
112×
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0×
1×
207×
180×
358×
350×
338×
322×
302×
278×
250×
218×
182×
142×
98×
50×
0s:
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0×
1×
25×
0×
252×
582×
2,340×
1,709×
3,024×
9,360×
2,744×
2,179×
873×
662×
0s:
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0×
1×
25×
518×
182×
3,204×
5,566×
6,584×
12,782×
17,154×
21,762×
17,894×
14,032×
11,572×
9,206×
7,778×
6,108×
4,582×
3,198×
358×
0s:
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0×
1×
711×
1,044×
856×
3,404×
2,194×
671×
856×
2,744×
5,159×
7,519×
8,437×
10,138×
15,027×
10,895×
9,789×
9,800×
14,544×
12,151×
12,398×
10,997×
6,388×
3,886×
4,998×
5,521×
4,683×
4,201×
4,782×
5,086×
11,175×
9,145×
8,730×
9,145×
10,386×
9,634×
9,148×
13,590×
12,364×
14,644×
21,336×
17,000×
16,968×
16,182×
18,085×
15,480×
16,225×
10,653×
15,703×
17,574×
25,209×
18,843×
16,584×
12,139×
12,854×
17,404×
18,926×
14,804×
15,888×
18,736×
15,002×
13,880×
11,747×
6,700×
10,922×
7,447×
6,365×
3,250×
5,422×
5,366×
4,311×
6,883×
6,604×
5,912×
3,036×
0s:
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25s:
0×
1×
25×
0×
0×
10,248×
23,812×
50,962×
62,042×
68,336×
81,376×
100,594×
149,528×
230,614×
254,772×
317,776×
318,604×
349,766×
432,244×
339,658×
325,806×
155,028×
60,964×
22,924×
6,186×
4,590×
0s:
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0×
1×
25×
2,124×
10,476×
9,362×
12,535×
23,587×
34,798×
43,020×
58,065×
38,009×
38,417×
44,954×
54,250×
64,266×
35,273×
38,533×
39,451×
73,183×
79,227×
83,758×
106,904×
125,666×
142,902×
108,054×
187,930×
222,696×
196,247×
134,039×
119,189×
93,753×
76,405×
70,044×
60,071×
60,250×
33,923×
27,114×
33,648×
28,032×
39,516×
52,992×
26,993×
33,722×
24,576×
72,006×
173,315×
196,668×
251,681×
236,478×
211,650×
222,263×
270,412×
263,085×
263,803×
256,996×
234,440×
257,882×
296,469×
279,521×
240,036×
233,014×
225,539×
187,025×
175,397×
201,726×
219,226×
239,008×
224,020×
277,817×
291,287×
226,002×
199,424×
218,393×
231,779×
235,845×
229,657×
187,184×
198,515×
197,569×
184,254×
173,852×
154,974×
112,496×
93,818×
95,307×
89,565×
73,215×
77,275×
68,407×
70,833×
80,070×
68,809×
70,347×
92,704×
99,350×
86,402×
95,697×
145,445×
142,349×
109,920×
85,825×
64,216×
57,648×
54,832×
41,392×
54,953×
34,679×
40,338×
62,648×
68,342×
36,888×
47,803×
42,875×
28,411×
22,670×
24,092×
18,047×
25,763×
29,040×
48,304×
32,275×
6,974×
13,739×
7,630×
6,452×
9,967×
8,833×
4,653×
6,666×
0s:
1s:
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47s:
0×
1×
7,767×
47,570×
105,744×
160,398×
212,822×
289,310×
240,776×
396,952×
413,070×
796,018×
1,334,030×
2,327,052×
3,128,586×
3,642,000×
2,915,082×
1,746,096×
1,806,198×
2,146,254×
2,647,664×
2,362,178×
1,919,866×
1,623,596×
1,687,806×
1,678,552×
1,788,686×
2,246,868×
2,089,314×
1,540,348×
905,972×
985,050×
1,150,858×
1,135,258×
1,037,928×
776,058×
708,926×
501,202×
542,870×
711,852×
787,400×
868,098×
646,114×
253,892×
65,304×
43,556×
16,520×
13,794×
0s:
1s:
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250s:
0×
1×
4,369×
51,526×
75,619×
55,048×
43,050×
19,104×
75,691×
133,019×
277,471×
410,969×
408,993×
622,365×
652,311×
481,672×
301,106×
172,973×
184,840×
183,330×
189,092×
194,572×
212,181×
184,292×
72,106×
108,282×
79,811×
72,111×
70,391×
77,460×
51,057×
16,622×
1,808×
4,868×
15,106×
7,030×
7,604×
54,738×
425,884×
524,239×
557,355×
499,312×
294,948×
395,769×
246,065×
218,265×
122,520×
134,454×
191,538×
283,014×
295,834×
299,322×
213,808×
182,297×
314,184×
450,895×
679,323×
616,315×
705,347×
638,292×
1,072,978×
1,400,207×
1,374,476×
1,254,542×
706,292×
667,542×
619,176×
621,722×
515,555×
490,999×
360,392×
399,278×
330,886×
319,276×
417,475×
380,318×
324,609×
240,848×
164,525×
174,486×
182,369×
254,231×
161,753×
69,354×
107,725×
64,542×
48,879×
15,236×
5,256×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
192,043×
1,645,737×
2,358,486×
3,180,652×
3,742,897×
3,385,356×
2,733,505×
2,433,357×
2,691,837×
2,539,720×
2,872,629×
2,977,648×
3,123,085×
3,019,587×
2,450,451×
2,166,102×
2,017,193×
2,041,538×
1,946,645×
1,678,829×
1,629,115×
1,795,318×
2,309,603×
2,581,852×
3,009,146×
2,846,468×
3,208,474×
2,927,733×
2,451,395×
2,292,655×
2,364,996×
2,393,458×
2,434,234×
2,231,109×
2,417,813×
2,392,920×
2,573,714×
1,905,749×
1,889,332×
1,613,725×
1,577,503×
983,821×
1,010,386×
1,234,511×
1,090,015×
1,072,439×
867,535×
903,941×
1,282,952×
1,050,344×
1,173,389×
1,070,489×
1,045,179×
1,201,921×
1,526,839×
1,585,067×
1,263,024×
1,096,697×
1,058,442×
1,293,430×
1,095,132×
878,967×
713,969×
814,948×
878,706×
1,096,779×
1,409,450×
1,433,003×
1,533,015×
1,657,622×
1,643,170×
1,432,285×
1,423,564×
1,667,898×
1,758,116×
1,575,434×
1,353,330×
1,172,164×
1,183,357×
1,015,580×
549,292×
407,550×
317,409×
353,120×
520,797×
843,553×
1,093,151×
824,824×
514,269×
296,266×
282,541×
344,118×
404,767×
248,799×
180,829×
156,870×
159,409×
82,571×
179,841×
106,850×
290,539×
293,816×
152,714×
105,046×
135,782×
249,641×
329,481×
223,718×
134,572×
123,710×
113,898×
130,459×
177,685×
250,195×
122,516×
92,385×
93,982×
95,790×
72,076×
101,480×
60,321×
36,755×
66,808×
26,106×
25,882×
10,199×
0s:
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0×
1×
25×
0×
0×
0×
158,268×
1,119,926×
3,216,140×
6,257,728×
10,264,472×
13,873,546×
17,024,928×
19,205,542×
19,083,726×
19,689,182×
17,019,772×
14,649,142×
15,924,108×
14,464,984×
16,401,840×
18,143,718×
22,339,328×
25,740,518×
27,182,492×
27,698,798×
29,226,886×
32,314,080×
32,338,742×
31,329,468×
25,337,228×
18,063,800×
14,381,450×
10,275,900×
6,893,676×
5,464,168×
5,567,102×
7,301,498×
8,778,374×
13,116,268×
13,385,554×
10,379,240×
4,771,394×
2,489,512×
2,000,406×
3,304,390×
2,773,038×
1,816,436×
1,600,210×
1,667,430×
1,420,860×
1,066,808×
1,595,814×
1,796,164×
670,588×
173,818×
46,810×
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0×
1×
207×
18,684×
384,784×
826,214×
714,211×
352,560×
183,508×
260,805×
176,060×
143,820×
1,314,867×
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1,273,329×
1,298,930×
1,729,191×
1,353,079×
478,673×
409,206×
139,143×
21,467×
5,970×
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405,114×
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1,727,110×
913,792×
538,847×
444,015×
443,748×
933,773×
1,134,142×
798,442×
599,995×
620,408×
796,444×
676,327×
643,515×
508,128×
555,968×
569,273×
556,694×
639,749×
937,459×
2,145,055×
1,717,550×
1,377,891×
780,420×
249,026×
150,601×
133,977×
54,250×
111,872×
79,597×
45,722×
13,482×
4,426×
1,424×
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0×
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288,720×
3,776,554×
12,944,170×
24,687,406×
30,415,592×
46,551,680×
59,134,646×
67,194,524×
69,926,762×
69,598,536×
76,417,572×
98,184,576×
130,436,414×
152,554,566×
183,568,276×
203,848,894×
257,394,790×
302,836,788×
310,310,116×
320,486,618×
330,394,090×
340,761,760×
313,505,966×
276,945,092×
256,255,456×
190,734,496×
163,052,394×
131,717,224×
109,869,080×
94,413,654×
84,251,812×
70,419,990×
69,910,854×
63,448,862×
64,573,592×
61,480,484×
57,672,004×
58,839,672×
62,191,772×
48,673,222×
36,276,906×
24,635,594×
17,874,194×
16,626,368×
18,370,432×
14,159,716×
10,425,014×
3,512,354×
1,926,526×
906,562×
554,008×
13,920×