Base 66: Iteration Frequencies: Cycle Start

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
Cycle Start		0.98 step
avg		1.47 step
avg		14.01 step
avg		2.94 step
avg		15.99 step
avg		51.18 step
avg		17.93 step
avg		55.17 step
avg
Step Frequency
Cycle Start
0s:
1s:
1×
65×
0s:
1s:
2s:
18×
1,137×
1,056×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
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16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
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:
1×
51,745×
138,480×
554,400×
59,335×
51,395×
5,193×
3,952×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
1×
15,001×
18,286×
28,546×
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:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
56s:
57s:
58s:
59s:
60s:
61s:
62s:
63s:
64s:
65s:
66s:
67s:
68s:
69s:
70s:
71s:
72s:
73s:
74s:
75s:
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:
145s:
146s:
147s:
148s:
149s:
150s:
151s:
152s:
153s:
154s:
155s:
156s:
157s:
158s:
159s:
160s:
161s:
162s:
163s:
164s:
165s:
166s:
167s:
168s:
169s:
170s:
171s:
172s:
1×
419,322×
844,565×
1,082,274×
857,297×
1,055,948×
1,088,024×
1,407,990×
1,472,836×
1,429,774×
1,616,184×
1,575,244×
1,426,114×
1,588,691×
1,727,721×
2,080,622×
2,019,123×
1,844,497×
2,048,005×
2,046,570×
2,055,809×
2,131,940×
2,137,762×
2,438,116×
2,249,577×
2,443,995×
2,138,599×
2,116,961×
1,806,716×
1,708,610×
1,763,600×
1,572,799×
1,451,270×
1,574,400×
1,470,293×
1,742,721×
1,479,731×
1,406,412×
1,670,650×
1,787,350×
1,517,274×
1,417,413×
1,245,028×
1,353,876×
1,196,225×
1,340,317×
1,435,696×
1,715,216×
1,589,049×
1,425,418×
1,283,945×
1,315,263×
1,271,275×
1,385,690×
1,416,801×
1,484,144×
1,302,285×
1,224,608×
1,156,739×
1,236,373×
1,203,158×
1,141,245×
1,419,270×
1,509,206×
1,262,238×
1,025,623×
1,195,223×
1,183,400×
1,422,227×
1,471,004×
1,327,795×
1,261,792×
924,457×
851,520×
949,794×
740,776×
713,699×
831,036×
842,732×
968,834×
1,198,935×
1,135,923×
946,329×
990,272×
878,164×
921,556×
925,986×
886,310×
1,049,773×
851,043×
662,817×
648,118×
514,785×
503,236×
494,786×
442,570×
454,272×
548,920×
497,845×
453,507×
469,605×
427,149×
422,344×
446,729×
315,646×
258,282×
291,032×
320,808×
290,873×
278,401×
372,708×
346,703×
365,617×
284,533×
274,436×
358,778×
312,659×
382,899×
382,348×
219,716×
177,837×
122,574×
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:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
1×
349,483×
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:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
56s:
57s:
58s:
59s:
60s:
61s:
62s:
63s:
64s:
65s:
66s:
67s:
68s:
69s:
70s:
71s:
72s:
73s:
74s:
75s:
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:
145s:
146s:
147s:
148s:
149s:
150s:
151s:
152s:
153s:
154s:
155s:
156s:
157s:
158s:
159s:
160s:
161s:
1×
6,455,356×
26,676,726×
19,170,259×
27,736,410×
30,924,332×
34,593,813×
39,852,179×
53,526,501×
53,288,825×
63,620,300×
71,017,514×
81,710,951×
90,205,737×
100,669,182×
97,843,148×
110,703,900×
113,484,224×
116,618,172×
122,572,832×
131,884,668×
129,111,741×
127,985,754×
134,707,477×
136,947,374×
152,236,846×
148,083,381×
146,817,738×
149,974,467×
147,032,301×
156,896,272×
168,384,009×
155,708,873×
155,925,179×
155,807,934×
156,078,301×
154,371,918×
167,995,174×
161,537,651×
168,269,454×
160,238,259×
169,648,646×
163,135,713×
170,929,636×
171,352,131×
181,244,565×
192,305,844×
187,634,924×
185,132,225×
184,988,673×
176,565,310×
179,586,860×
182,464,060×
205,915,297×
200,582,317×
193,298,076×
179,648,753×
192,033,728×
174,701,508×
187,575,295×
174,535,945×
172,360,948×
169,156,368×
171,645,522×
175,583,908×
182,588,664×
176,616,302×
181,931,600×
176,352,076×
168,136,726×
152,314,601×
145,052,844×
138,455,911×
123,323,071×
110,445,487×
115,080,504×
107,291,642×
107,664,066×
104,060,469×
97,335,492×
102,937,143×
108,695,612×
106,078,804×
105,290,114×
97,041,476×
94,593,814×
95,075,837×
87,954,717×
86,444,726×
87,704,808×
86,370,727×
77,003,152×
68,319,939×
68,901,702×
66,315,434×
65,907,387×
65,780,843×
63,285,123×
57,994,486×
55,681,666×
51,370,149×
47,716,589×
47,658,484×
51,175,662×
44,586,914×
42,387,995×
36,294,549×
40,206,684×
38,433,878×
39,169,077×
30,998,866×
29,398,115×
28,621,502×
25,903,357×
24,302,255×
20,126,462×
21,238,878×
21,113,892×
20,519,231×
19,016,724×
15,020,599×
13,100,703×
11,132,778×
12,582,280×
15,140,704×
16,808,130×
14,253,302×
13,364,069×
13,299,049×
13,554,095×
11,219,051×
13,565,371×
9,235,828×
8,971,995×
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×