Base 13: Iteration Frequencies: Full Cycle

Radix Character Encoding
0	1	2	3	4	5	6	7	8	9	A	B	C

Color Key
Orange	Fixed Point	number that becomes itself after one iteration
Green	Final Numbers	all numbers eventually reached, from all cycles
Purple	Cycle Start	iterations include entry into cycle, assuming it repeats
Red	Full Cycle	iterations include entire cycle, proving it repeats
Blue	Start Numbers	subset (optimization) of all numbers; minimal for full coverage
Gray	No Results	any calculation that has no results

Digits

Average Steps
Full Cycle

1.92 step
avg

4.57 step
avg

5.35 step
avg

7.46 step
avg

10.65 step
avg

12.84 step
avg

12.22 step
avg

10.96 step
avg

17.98 step
avg

13.24 step
avg

28.75 step
avg

19.43 step
avg

29.56 step
avg

22.40 step
avg

36.03 step
avg

28.62 step
avg

36.82 step
avg

26.40 step
avg

29.67 step
avg

21.34 step
avg

28.20 step
avg

23.65 step
avg

17.56 step
avg

33.25 step
avg

18.59 step
avg

26.79 step
avg

15.44 step
avg

37.27 step
avg

15.57 step
avg

Step Frequency
Full Cycle

0s:
1s:
2s:
0×
1×
12×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
0×
1×
12×
8×
19×
15×
36×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
0×
1×
12×
97×
45×
88×
78×
64×
46×
24×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
0×
1×
12×
0×
70×
165×
101×
280×
878×
265×
48×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
0×
1×
12×
0×
0×
457×
495×
305×
747×
431×
299×
430×
489×
941×
1,066×
373×
142×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
0×
1×
92×
140×
670×
487×
321×
757×
605×
914×
915×
1,191×
1,422×
1,950×
2,502×
1,533×
1,109×
1,600×
909×
833×
334×
279×

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:
0×
1×
12×
816×
1,152×
1,961×
2,954×
2,838×
3,005×
5,667×
4,984×
3,494×
1,253×
2,889×
3,746×
3,054×
1,940×
1,658×
2,187×
1,201×
749×
760×
1,053×
1,464×
1,000×
385×
165×

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:
0×
1×
132×
564×
3,963×
5,435×
7,531×
9,308×
10,327×
12,864×
11,489×
12,195×
10,659×
11,669×
9,419×
5,161×
3,440×
2,007×
2,312×
1,474×
1,180×
1,806×
1,273×
1,347×
400×
14×

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:
0×
1×
12×
0×
0×
0×
4,981×
10,229×
14,171×
11,971×
7,542×
8,938×
10,174×
11,025×
12,731×
18,449×
18,966×
16,785×
12,585×
16,043×
17,078×
14,806×
13,747×
12,428×
7,813×
9,307×
13,386×
7,938×
6,515×
4,371×
1,792×
740×
903×
1,016×
1,005×
565×
760×
1,424×
2,066×
1,667×

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:
0×
1×
132×
1,076×
9,499×
17,450×
16,783×
23,025×
31,045×
48,300×
40,332×
56,294×
60,455×
57,957×
44,757×
45,926×
39,971×
30,742×
23,723×
26,728×
23,034×
17,042×
12,485×
6,744×
5,559×
4,575×
2,303×
300×
408×

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:
0×
1×
12×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
5,564×
20,728×
37,147×
44,313×
60,311×
63,841×
70,065×
54,893×
77,453×
98,433×
108,557×
82,367×
64,610×
57,727×
53,192×
39,857×
27,861×
22,064×
36,504×
15,976×
12,172×
12,276×
14,803×
14,700×
8,387×
15,688×
32,076×
38,589×
32,959×
24,385×
17,693×
13,922×
9,815×
6,122×
5,932×
7,973×
8,760×
10,097×
9,970×
9,700×
3,696×
635×
252×

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:
0×
1×
132×
2,308×
20,185×
42,217×
34,089×
21,924×
36,327×
92,624×
62,006×
40,509×
42,604×
79,288×
92,501×
107,362×
135,566×
160,650×
152,094×
171,983×
216,742×
185,762×
185,798×
144,597×
105,550×
88,385×
109,890×
88,679×
51,923×
43,904×
63,734×
50,237×
22,687×
21,013×
15,218×
4,995×
2,502×
3,670×
4,500×

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:
0×
1×
996×
8,484×
49,686×
97,959×
61,652×
55,206×
166,042×
282,657×
144,151×
116,582×
69,042×
34,013×
13,888×
5,003×
9,443×
92,080×
345,041×
667,772×
653,742×
653,838×
722,084×
737,931×
827,711×
729,016×
624,720×
582,867×
768,635×
403,110×
255,243×
232,186×
109,147×
30,573×
37,278×
30,109×
13,656×
8,377×
6,646×
1,848×
1,835×
4,550×
1,568×
280×
165×
779×
108×

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:
0×
1×
12×
1,020×
11,028×
30,064×
24,004×
9,364×
4,684×
1,676×
213×
0×
0×
0×
0×
12,336×
97,378×
233,101×
373,838×
517,604×
464,749×
317,716×
323,644×
402,526×
432,857×
563,164×
815,777×
876,895×
925,640×
631,405×
431,534×
299,289×
259,268×
222,179×
204,860×
213,971×
157,374×
215,142×
221,406×
262,986×
361,201×
465,464×
514,202×
625,074×
595,280×
645,479×
534,626×
551,901×
584,288×
358,476×
367,438×
264,670×
295,613×
374,420×
275,400×
250,524×
233,505×
201,988×
107,252×
92,614×
67,492×
26,108×
15,305×
11,441×
2,709×
765×
1,920×

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:
0×
1×
132×
2,612×
50,372×
198,669×
129,172×
207,803×
424,201×
799,664×
415,869×
455,481×
769,732×
618,564×
856,054×
1,359,712×
855,995×
664,787×
713,206×
530,273×
408,684×
327,334×
718,048×
596,901×
831,898×
1,346,365×
1,303,292×
1,526,395×
1,522,392×
1,165,891×
604,897×
789,042×
446,106×
423,141×
560,335×
179,786×
89,086×
63,605×
20,447×
54,244×
414,453×
511,049×
485,526×
786,186×
880,345×
1,176,875×
570,148×
464,324×
719,721×
482,192×
235,582×
190,705×
73,022×
31,099×
20,489×
195,591×
440,966×
135,641×
79,906×
197,349×
126,359×
120,611×
38,792×
6,362×
6,302×
1,924×
48×

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:
0×
1×
12×
1,740×
17,680×
41,472×
29,838×
10,320×
6,404×
0×
0×
0×
0×
0×
0×
0×
0×
0×
15,916×
162,848×
347,453×
513,627×
556,737×
484,834×
393,112×
553,341×
1,157,392×
1,916,472×
2,236,434×
2,322,136×
2,786,544×
3,026,192×
3,136,784×
3,235,374×
3,425,059×
2,865,322×
2,783,887×
2,048,994×
1,457,263×
1,356,075×
923,968×
1,156,571×
669,211×
853,623×
883,361×
917,472×
885,405×
568,907×
506,150×
469,712×
427,068×
642,786×
633,811×
846,864×
663,315×
803,905×
766,831×
396,093×
264,063×
180,738×
244,187×
239,576×
187,061×
172,784×
240,475×
222,743×
136,012×
72,802×
26,086×
3,624×
1,468×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
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28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
0×
1×
132×
3,124×
74,941×
366,805×
201,735×
191,283×
735,389×
1,245,390×
591,925×
486,317×
530,539×
277,714×
315,359×
859,405×
2,671,576×
3,128,137×
2,219,686×
3,501,089×
4,551,783×
4,886,544×
2,884,652×
2,417,764×
2,373,991×
1,637,198×
1,789,306×
4,683,545×
4,839,815×
6,298,856×
5,334,647×
4,588,561×
3,292,133×
3,660,495×
3,040,050×
2,194,809×
1,524,895×
2,476,187×
2,676,450×
1,230,265×
485,621×
720,215×
500,477×
394,822×
461,314×
91,687×
33,789×
12,374×
2,681×
1,884×
4,812×
1,056×

0s:
1s:
2s:
3s:
4s:
5s:
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40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
0×
1×
12×
1,020×
17,264×
67,528×
47,158×
14,928×
5,884×
12,716×
231,969×
853,680×
2,038,346×
2,603,523×
4,388,893×
3,306,278×
4,000,270×
3,436,403×
3,583,342×
3,897,314×
4,400,282×
4,322,932×
5,374,374×
4,419,668×
4,159,589×
3,951,427×
3,090,750×
2,335,077×
2,012,083×
1,247,468×
1,148,991×
1,581,280×
3,502,568×
4,427,175×
7,024,150×
6,178,393×
6,340,994×
7,655,799×
6,513,267×
8,073,739×
7,451,394×
5,755,562×
4,246,283×
3,068,420×
1,874,117×
1,072,256×
726,438×
497,775×
128,073×
30,480×
3,155×
37×

0s:
1s:
2s:
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31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
0×
1×
1,860×
47,952×
256,635×
812,023×
1,378,234×
3,421,692×
3,535,212×
6,313,164×
5,035,215×
7,420,363×
7,632,527×
3,898,636×
3,525,918×
6,786,743×
11,169,344×
12,550,426×
10,896,900×
12,670,716×
12,895,536×
12,642,675×
8,084,129×
5,969,831×
6,362,424×
5,575,983×
5,068,307×
10,112,277×
9,027,016×
9,952,808×
10,569,722×
12,197,510×
7,395,634×
5,141,169×
3,060,171×
1,493,234×
1,125,635×
1,148,083×
356,836×
242,343×
17,842×
114×

0s:
1s:
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0×
1×
12×
1,020×
24,912×
164,984×
284,435×
69,460×
23,257×
405,064×
1,874,257×
6,914,748×
8,350,505×
8,778,319×
6,165,242×
4,676,885×
5,640,734×
3,825,721×
2,307,261×
1,872,541×
1,664,313×
4,803,781×
14,125,694×
17,990,119×
16,824,641×
18,573,012×
19,926,626×
18,897,232×
20,366,391×
17,769,714×
20,710,640×
16,115,591×
14,540,952×
13,592,797×
8,463,397×
6,988,229×
6,013,873×
9,064,972×
9,534,268×
9,178,440×
8,793,709×
6,768,714×
7,356,812×
5,888,837×
3,708,532×
2,381,629×
1,749,073×
991,515×
485,488×
128,650×
34,224×
6,097×

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0×
1×
132×
4,868×
151,388×
1,017,436×
397,511×
526,268×
2,094,691×
4,245,098×
4,715,531×
12,285,086×
11,669,662×
18,139,147×
12,808,225×
10,600,225×
17,073,750×
20,246,757×
25,365,644×
26,527,301×
29,221,664×
35,751,809×
21,583,274×
20,121,802×
25,968,045×
21,499,594×
15,620,512×
20,686,094×
16,849,721×
25,608,232×
26,689,815×
26,512,567×
29,711,974×
22,893,521×
14,575,173×
11,275,689×
6,008,393×
4,872,623×
2,915,891×
882,599×
333,547×
468,397×
247,198×
15,798×
26,154×
120,989×
23,044×
1,200×

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0×
1×
3,084×
177,604×
1,036,148×
4,147,656×
6,988,375×
7,226,120×
9,345,684×
16,706,178×
24,063,661×
36,174,069×
46,680,547×
53,286,414×
58,355,416×
53,235,249×
50,768,889×
44,095,772×
45,091,617×
52,205,126×
56,507,930×
51,605,966×
56,112,482×
47,178,467×
37,131,474×
28,729,224×
19,606,938×
12,126,749×
7,773,746×
4,001,755×
2,260,599×
1,197,551×
385,859×
201,476×
40,089×
3,885×

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0×
1×
132×
4,660×
189,694×
1,587,245×
647,337×
674,121×
3,655,046×
6,564,993×
2,286,913×
3,568,439×
10,211,110×
12,395,594×
8,668,824×
6,530,202×
22,378,643×
27,740,290×
27,780,705×
50,889,416×
49,410,674×
50,187,671×
35,570,245×
20,215,101×
27,119,444×
30,052,947×
24,846,678×
42,120,660×
36,610,215×
47,918,064×
52,748,772×
42,714,133×
40,553,198×
35,616,954×
21,641,855×
10,056,002×
8,660,006×
7,861,149×
8,507,487×
11,539,948×
9,821,739×
18,705,107×
34,365,384×
50,729,355×
38,717,573×
29,827,374×
27,841,039×
22,651,364×
32,133,595×
49,240,531×
33,077,780×
26,063,050×
23,251,250×
19,457,037×
14,591,046×
10,399,477×
5,867,935×
6,784,810×
5,014,962×
2,074,644×
873,089×
252,708×
166,039×
46,244×

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0×
1×
7,180×
452,848×
3,100,496×
14,601,796×
24,432,101×
25,066,957×
31,839,466×
27,448,786×
26,100,267×
58,314,871×
75,846,455×
82,708,261×
101,053,706×
109,092,795×
110,039,997×
121,821,542×
115,365,838×
103,894,270×
88,440,681×
95,535,941×
103,389,731×
88,335,484×
113,224,351×
93,993,942×
85,050,842×
63,926,671×
37,280,146×
23,871,737×
10,419,946×
7,153,133×
6,696,843×
2,200,888×
819,580×
511,158×
267,184×
103,053×
61,904×
10,532×
1,616×

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0×
1×
132×
8,004×
389,935×
4,734,571×
15,986,493×
45,301,813×
47,520,531×
59,805,603×
65,952,464×
67,969,541×
54,617,659×
38,251,727×
52,393,978×
47,567,806×
69,399,507×
71,560,361×
79,351,350×
95,410,803×
120,388,800×
110,550,434×
70,819,034×
57,450,983×
75,315,376×
83,896,294×
58,475,369×
63,684,202×
70,683,129×
116,238,368×
103,354,627×
89,477,920×
63,957,152×
59,211,800×
45,436,437×
46,476,991×
59,613,367×
41,397,328×
35,791,183×
36,851,242×
37,130,015×
71,125,055×
51,052,705×
28,042,295×
37,898,363×
39,508,623×
42,056,625×
39,899,439×
36,659,017×
21,688,496×
11,292,423×
11,821,840×
6,799,586×
7,330,808×
12,536,065×
11,163,441×
4,757,040×
2,208,188×
4,429,918×
2,502,267×
1,167,599×
783,080×
273,529×
32,133×
23,428×
855×

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0×
1×
11,276×
803,596×
6,108,476×
34,277,400×
64,430,676×
72,264,475×
111,181,249×
136,022,104×
147,449,880×
219,462,371×
248,309,462×
284,386,602×
351,293,832×
351,700,924×
342,354,701×
311,023,751×
272,285,994×
212,601,081×
178,434,733×
142,849,764×
117,577,524×
105,627,322×
73,696,949×
41,931,905×
28,174,119×
24,227,474×
14,799,164×
7,677,157×
5,321,576×
3,114,411×
950,238×
296,651×
136,114×
13,617×
867×

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0×
1×
996×
50,564×
323,921×
3,439,227×
1,008,719×
1,337,474×
9,302,267×
16,175,473×
10,168,791×
46,209,511×
62,860,107×
44,767,134×
34,171,275×
25,956,344×
69,315,220×
76,696,637×
42,138,150×
83,985,044×
106,345,107×
108,725,843×
63,225,128×
41,396,468×
40,925,722×
30,759,120×
29,392,348×
123,073,111×
134,288,368×
158,426,012×
137,163,620×
133,323,694×
160,311,661×
173,265,965×
155,985,631×
194,921,382×
274,928,231×
282,641,768×
223,010,558×
163,019,413×
131,681,078×
96,199,449×
64,690,396×
131,344,945×
210,073,913×
181,382,525×
183,307,339×
181,339,336×
188,630,200×
151,502,719×
120,275,200×
111,785,655×
96,990,172×
71,687,580×
46,829,186×
30,499,771×
15,937,565×
29,443,809×
18,313,269×
19,707,667×
36,915,347×
46,676,068×
57,890,983×
49,601,045×
29,426,897×
12,230,567×
5,925,108×
1,997,214×
1,180,343×
306,338×
38,099×
6,589×
956×
147×

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0×
1×
17,676×
1,421,424×
11,633,200×
74,119,548×
154,600,370×
219,512,884×
372,939,496×
422,729,346×
471,047,621×
459,474,078×
424,077,428×
486,762,081×
417,924,474×
533,315,632×
533,595,051×
520,595,033×
443,839,298×
384,086,585×
346,595,014×
280,578,675×
295,246,623×
260,252,587×
218,282,766×
196,779,924×
150,642,158×
85,416,059×
65,811,180×
31,426,635×
11,648,382×
6,701,611×
3,952,381×
6,085,581×
3,740,260×
2,582,556×
896,841×
298,354×
9,363×
14,823×
1,921×