Base 31: 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
Average Steps
Cycle Start
0.97 step
avg
4.90 step
avg
6.89 step
avg
1.88 step
avg
7.41 step
avg
11.62 step
avg
12.80 step
avg
27.44 step
avg
11.63 step
avg
53.25 step
avg
33.64 step
avg
Step Frequency
Cycle Start
0s:
1s:

30×
0s:
1s:
2s:
3s:
4s:
5s:
6s:

30×
15×
30×
60×
120×
240×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:

541×
252×
502×
492×
478×
460×
438×
412×
382×
348×
310×
268×
222×
172×
118×
60×
0s:
1s:
2s:
3s:
4s:

10,670×
30,905×
4,545×
255×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:

9,124×
13,011×
10,452×
22,616×
31,804×
40,434×
37,365×
38,326×
39,670×
27,528×
23,605×
17,115×
9,264×
3,554×
763×
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:

14,678×
36,854×
62,486×
100,536×
141,149×
146,715×
179,349×
89,391×
77,238×
93,880×
144,924×
137,401×
91,020×
137,388×
73,882×
46,671×
49,713×
50,819×
36,978×
40,502×
17,104×
7,508×
13,602×
9,814×
16,444×
12,562×
18,752×
16,445×
22,511×
50,571×
7,910×
2,067×
578×
318×
31×
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:

38,602×
95,035×
195,390×
300,375×
392,443×
460,508×
543,485×
610,122×
829,312×
915,890×
959,963×
817,929×
742,876×
500,381×
323,335×
202,261×
205,041×
113,806×
120,149×
148,307×
179,163×
246,731×
244,528×
246,737×
229,509×
213,568×
163,387×
124,887×
72,219×
49,754×
9,778×
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:

51,509×
214,275×
756,096×
678,880×
673,125×
796,609×
972,730×
964,305×
1,031,148×
1,163,212×
1,149,693×
1,020,373×
1,174,830×
1,198,294×
1,237,451×
1,228,885×
1,110,345×
1,340,901×
1,618,554×
1,284,296×
963,752×
774,214×
1,119,373×
1,226,586×
1,250,545×
962,826×
1,212,649×
1,053,951×
978,001×
834,579×
1,024,938×
1,366,295×
1,493,882×
1,244,018×
1,117,617×
971,400×
1,219,861×
1,040,658×
619,999×
436,737×
454,994×
519,956×
492,946×
502,413×
452,050×
322,073×
270,770×
154,013×
118,245×
86,717×
135,627×
77,200×
58,456×
29,774×
97,599×
198,790×
298,324×
403,940×
268,346×
238,881×
294,996×
293,536×
130,363×
113,912×
246,393×
206,272×
203,377×
254,875×
248,254×
196,302×
209,709×
175,632×
94,628×
84,727×
68,553×
36,087×
38,809×
63,319×
147,954×
26,895×
10,392×
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:

439,421×
1,765,236×
4,014,763×
6,802,071×
11,131,893×
14,734,296×
16,770,804×
16,780,465×
17,343,488×
16,315,508×
14,892,461×
13,306,335×
12,280,489×
11,591,601×
11,094,638×
9,660,384×
7,103,182×
5,187,430×
3,230,898×
2,091,350×
1,101,438×
922,625×
1,001,758×
2,042,793×
1,409,613×
1,592,992×
925,753×
581,292×
533,915×
826,328×
1,037,111×
759,632×
583,587×
745,719×
754,276×
264,438×
244,720×
44,828×
5,600×
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:

50,022×
528,728×
4,620,079×
6,670,857×
6,267,550×
5,371,101×
7,177,159×
7,076,020×
6,255,763×
6,130,232×
7,623,967×
8,825,772×
8,041,595×
6,655,668×
6,265,561×
5,798,379×
5,503,628×
5,597,984×
3,781,527×
4,141,492×
5,171,901×
8,342,036×
7,631,515×
9,697,012×
9,772,743×
7,896,481×
7,492,681×
9,198,146×
10,724,041×
9,864,878×
8,253,247×
7,975,467×
7,709,075×
7,579,064×
7,231,427×
4,647,622×
3,027,119×
3,427,761×
5,269,313×
4,569,531×
5,194,840×
6,625,810×
7,202,109×
7,664,505×
8,590,684×
7,909,842×
7,759,925×
8,915,828×
12,191,222×
17,267,427×
18,945,710×
17,855,336×
16,396,888×
17,602,248×
18,988,919×
20,683,227×
16,827,093×
13,134,020×
13,087,372×
13,436,122×
15,375,030×
15,027,552×
13,713,263×
12,914,005×
10,565,786×
11,412,585×
8,783,753×
9,125,338×
9,878,176×
10,519,710×
9,990,690×
8,659,906×
6,338,015×
7,660,452×
9,754,937×
13,125,557×
10,189,002×
10,294,289×
7,332,280×
7,531,792×
6,719,158×
7,346,200×
6,710,136×
6,393,481×
6,231,145×
4,232,159×
2,885,362×
3,768,467×
4,736,383×
4,609,608×
7,348,494×
9,377,442×
6,314,160×
4,749,569×
2,798,696×
2,582,990×
3,571,648×
4,285,785×
2,503,625×
1,961,305×
2,634,979×
1,262,168×
1,340,718×
1,469,866×
2,126,534×
2,339,145×
2,127,387×
1,219,639×
1,000,519×
945,229×
410,956×
420,534×
587,004×
611,994×
528,819×
433,784×
197,536×
288,436×
178,000×
292,579×
543,922×
717,885×
1,673,016×
685,335×
350,532×
550,948×
534,390×
194,538×
140,933×
104,937×
57,811×
85,969×
8,329×
75,518×
34,524×
24,912×
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:

1,668,549×
8,907,843×
20,563,295×
32,658,912×
50,290,994×
77,608,244×
90,565,324×
82,340,184×
66,450,640×
65,278,474×
53,987,028×
42,566,488×
43,118,865×
44,468,863×
40,252,718×
39,977,787×
42,185,049×
43,218,212×
45,703,623×
55,310,947×
77,130,221×
79,570,067×
89,691,219×
77,300,977×
68,625,281×
56,463,286×
51,764,840×
43,125,349×
41,533,238×
43,867,216×
45,016,154×
60,161,349×
77,787,069×
98,505,370×
73,986,050×
53,449,566×
43,050,682×
39,525,905×
31,552,914×
23,929,446×
16,672,921×
11,912,700×
13,476,746×
24,083,009×
32,905,272×
45,010,109×
38,301,154×
39,946,352×
37,415,214×
32,565,319×
33,542,923×
30,598,457×
21,957,042×
20,813,309×
18,571,368×
18,846,390×
24,599,077×
25,819,450×
32,505,247×
30,366,332×
25,354,347×
25,188,755×
26,931,870×
24,863,955×
28,148,348×
30,050,180×
31,547,444×
35,454,565×
45,955,244×
48,347,008×
43,476,910×
31,867,666×
23,172,512×
21,468,985×
17,109,145×
15,829,842×
8,074,850×
3,034,006×
493,838×
25,868×