Base 66: Iteration Frequencies: Full Cycle

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789ABCDEF
GHIJKLMNOPQRSTUV
WXYZabcdefghijkl
mnopqrstuvwxyzΑΒ
ΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣ

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		33.51 step
avg		15.01 step
avg		7.94 step
avg		16.99 step
avg		67.10 step
avg		21.93 step
avg		162.19 step
avg
Step Frequency
Full Cycle
0s:
1s:
2s:
0×
1×
65×
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0×
1×
65×
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×
17×
1,072×
1,056×
0s:
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0×
1×
1,187×
1,120×
2,238×
2,230×
2,218×
2,202×
2,182×
2,158×
2,130×
2,098×
2,062×
2,022×
1,978×
1,930×
1,878×
1,822×
1,762×
1,698×
1,630×
1,558×
1,482×
1,402×
1,318×
1,230×
1,138×
1,042×
942×
838×
730×
618×
502×
382×
258×
130×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
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9s:
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0×
1×
65×
0×
0×
0×
51,680×
138,480×
554,400×
59,335×
51,395×
5,193×
3,952×
0s:
1s:
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0×
1×
11,703×
20,462×
29,668×
62,826×
103,246×
180,680×
226,074×
334,982×
306,454×
576,342×
619,540×
795,286×
729,302×
919,236×
718,750×
933,674×
764,716×
869,160×
695,054×
696,634×
543,648×
538,738×
411,562×
332,264×
256,394×
200,064×
137,810×
66,260×
20,244×
2,240×
0s:
1s:
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0×
1×
1,187×
0×
12,168×
20,592×
54,090×
93,692×
187,518×
167,476×
257,624×
162,293×
153,630×
197,686×
253,389×
301,447×
291,740×
281,230×
359,581×
558,929×
664,731×
680,509×
838,759×
738,994×
897,787×
971,278×
1,266,122×
1,108,997×
1,240,741×
1,165,608×
1,418,741×
1,430,024×
1,180,839×
1,342,962×
1,114,000×
1,209,460×
1,422,448×
1,726,278×
1,590,348×
1,602,911×
1,548,159×
1,886,728×
1,944,106×
1,815,319×
1,934,176×
2,208,791×
2,125,747×
2,313,369×
1,895,169×
2,003,964×
1,787,751×
1,881,284×
2,027,206×
2,150,752×
2,291,706×
2,014,330×
1,882,802×
1,744,429×
1,628,544×
2,062,346×
2,123,135×
2,130,151×
1,977,053×
1,732,441×
1,522,473×
1,579,816×
1,708,653×
1,656,870×
1,608,931×
2,018,068×
2,077,898×
1,901,730×
1,893,476×
1,822,294×
1,780,387×
1,475,180×
1,367,394×
1,523,710×
1,278,078×
1,273,671×
1,267,943×
1,294,911×
1,260,630×
1,068,278×
1,038,371×
1,061,763×
1,162,099×
1,260,205×
1,149,012×
1,118,305×
1,043,611×
974,403×
1,009,819×
1,037,002×
1,010,887×
1,166,309×
1,147,902×
1,074,492×
1,024,214×
816,330×
896,808×
766,330×
797,183×
781,586×
910,529×
741,984×
631,773×
611,328×
575,759×
613,005×
602,322×
475,808×
370,352×
391,846×
342,340×
273,747×
253,627×
347,624×
362,771×
386,609×
300,973×
302,298×
385,122×
354,562×
422,607×
419,863×
270,503×
191,499×
131,460×
95,994×
113,664×
155,395×
177,780×
144,071×
68,740×
62,308×
49,920×
80,677×
132,470×
161,559×
186,873×
274,681×
170,132×
146,779×
160,794×
196,653×
265,385×
249,646×
184,948×
213,290×
274,949×
157,568×
133,768×
116,756×
175,402×
162,617×
173,125×
163,534×
125,466×
137,940×
99,868×
137,352×
146,761×
178,280×
115,651×
150,094×
205,490×
133,060×
114,752×
113,062×
82,890×
127,972×
82,198×
13,900×
22,912×
20,110×
22,726×
20,696×
20,400×
35,340×
0s:
1s:
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35s:
36s:
37s:
0×
1×
65×
0×
0×
349,418×
916,094×
2,309,688×
2,907,056×
4,713,102×
7,550,432×
11,474,934×
15,611,732×
23,565,780×
33,162,924×
41,635,306×
52,953,434×
67,385,336×
77,291,708×
83,197,910×
103,405,016×
116,266,580×
119,882,448×
123,197,096×
125,885,362×
111,813,262×
99,032,716×
85,048,168×
63,974,180×
39,273,164×
26,534,646×
17,799,236×
8,625,500×
4,040,134×
2,274,032×
880,476×
123,682×
29,086×
0s:
1s:
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0×
1×
22,329×
64,562×
0×
0×
121,648×
1,368,666×
4,044,450×
3,606,131×
5,985,177×
5,608,351×
6,889,089×
9,356,391×
12,563,975×
13,134,569×
18,056,586×
17,569,507×
23,991,658×
22,165,663×
28,137,846×
26,785,206×
38,474,379×
38,684,289×
42,072,030×
49,276,660×
54,998,109×
48,048,545×
44,165,681×
45,182,403×
45,618,906×
54,489,232×
55,969,081×
53,695,943×
54,696,105×
53,053,204×
55,376,129×
57,100,918×
60,227,711×
60,282,660×
64,311,358×
61,259,848×
61,263,279×
66,196,001×
68,576,730×
77,083,358×
75,827,486×
83,773,524×
78,683,319×
97,520,637×
96,545,464×
105,632,482×
109,347,251×
115,512,991×
114,084,724×
118,158,751×
108,663,920×
104,768,697×
102,566,582×
114,262,643×
105,762,633×
105,991,232×
103,122,830×
104,910,996×
103,253,482×
110,914,761×
105,775,913×
93,536,209×
90,967,368×
86,916,724×
87,297,286×
91,713,876×
91,783,517×
83,291,668×
77,855,307×
67,138,476×
63,879,209×
65,919,559×
66,191,060×
59,084,759×
49,171,343×
50,573,480×
45,209,004×
47,875,293×
43,655,606×
38,629,874×
37,533,394×
38,254,729×
36,202,161×
41,714,377×
37,425,511×
34,949,829×
30,970,352×
28,996,430×
25,069,339×
22,058,691×
22,570,215×
19,333,616×
19,699,358×
21,205,324×
22,263,525×
23,795,938×
19,653,764×
14,550,002×
13,332,296×
11,241,291×
8,700,766×
6,576,256×
5,741,325×
5,634,209×
5,223,081×
5,422,997×
4,685,017×
4,162,082×
5,114,904×
5,165,307×
3,604,267×
2,909,509×
4,735,990×
5,427,869×
7,583,798×
5,763,440×
5,839,562×
5,550,069×
2,668,753×
2,151,522×
2,472,527×
2,404,333×
1,716,283×
2,097,825×
2,222,672×
3,729,866×
3,789,446×
3,393,083×
2,889,601×
2,011,072×
790,374×
1,224,226×
1,315,736×
1,125,510×
1,222,108×
1,282,466×
516,072×
450,853×
633,342×
579,157×
345,784×
109,191×
444,841×
268,120×
316,420×
165,990×
254,608×
101,828×
76,200×
76,791×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
3,154,178×
15,040,414×
10,843,562×
13,010,961×
15,837,096×
17,979,773×
21,282,007×
27,293,147×
23,216,750×
28,702,499×
31,334,672×
34,922,012×
46,808,328×
46,368,841×
44,844,364×
45,951,444×
50,775,876×
54,613,435×
56,075,618×
56,445,054×
61,550,835×
69,577,693×
71,509,333×
71,771,752×
77,495,480×
70,287,734×
70,940,977×
80,861,676×
76,028,678×
87,918,196×
92,393,031×
83,306,496×
85,036,789×
82,045,828×
87,379,174×
86,917,922×
93,917,270×
89,859,123×
90,212,177×
86,139,557×
93,022,835×
91,457,475×
85,008,701×
79,026,271×
84,357,750×
94,143,211×
84,471,917×
80,796,859×
82,520,613×
82,130,423×
81,064,051×
88,386,890×
101,496,540×
101,303,702×
99,250,234×
94,415,757×
102,437,001×
90,232,720×
93,523,780×
90,623,432×
93,743,852×
90,281,853×
91,594,520×
94,860,641×
95,545,401×
94,292,416×
106,174,874×
106,794,368×
106,188,855×
93,370,592×
84,721,650×
79,668,528×
74,002,816×
69,369,917×
74,317,208×
68,072,124×
68,323,497×
67,470,792×
62,988,246×
66,827,612×
74,789,057×
74,566,655×
73,892,209×
70,009,949×
68,142,376×
73,731,331×
65,791,154×
65,708,664×
71,263,875×
65,701,891×
60,235,345×
56,555,638×
58,789,752×
55,957,417×
52,796,254×
53,215,322×
51,209,721×
47,420,354×
46,677,734×
45,124,427×
44,335,733×
43,869,673×
47,462,735×
40,897,326×
38,391,894×
33,024,633×
37,322,098×
34,465,040×
35,721,906×
28,492,270×
27,723,109×
26,370,029×
23,059,952×
20,230,583×
16,540,641×
18,126,298×
18,995,165×
19,097,968×
17,602,090×
13,929,432×
12,419,641×
10,675,745×
12,122,972×
14,571,208×
16,369,122×
13,945,558×
13,051,093×
12,982,632×
13,396,817×
11,119,413×
13,460,211×
9,133,576×
8,938,713×
8,973,122×
9,041,112×
8,015,063×
6,860,495×
5,529,803×
5,387,840×
5,611,252×
6,647,190×
5,645,135×
3,018,253×
2,601,425×
2,421,885×
1,592,360×
1,242,080×
476,436×
362,592×
94,406×
109,474×
83,803×
197,044×
249,350×
85,788×
269,546×
286,542×
693,914×
1,204,060×
1,036,160×
184,352×