Base 15: Iteration Frequencies: Full Cycle

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

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

1.93 step
avg

4.85 step
avg

5.76 step
avg

5.54 step
avg

8.20 step
avg

14.11 step
avg

15.69 step
avg

19.95 step
avg

27.93 step
avg

21.68 step
avg

18.93 step
avg

31.15 step
avg

39.10 step
avg

25.63 step
avg

42.92 step
avg

37.01 step
avg

24.52 step
avg

24.84 step
avg

20.84 step
avg

51.50 step
avg

22.33 step
avg

31.46 step
avg

17.35 step
avg

33.23 step
avg

Step Frequency
Full Cycle

0s:
1s:
2s:
0×
1×
14×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
0×
1×
14×
7×
14×
28×
56×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
0×
1×
14×
127×
60×
118×
108×
94×
76×
54×
28×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
0×
1×
56×
187×
412×
495×
1,276×
570×
63×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
0×
1×
194×
419×
773×
1,482×
1,522×
964×
688×
622×
1,132×
1,942×
1,204×
498×
187×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
0×
1×
14×
0×
215×
410×
905×
1,151×
1,404×
1,431×
1,022×
1,888×
3,017×
5,792×
5,178×
3,469×
2,062×
3,816×
870×
748×
1,902×
2,772×
624×
69×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
23s:
0×
1×
14×
0×
0×
0×
0×
0×
0×
2,531×
2,846×
5,588×
7,544×
12,604×
11,445×
13,039×
11,656×
12,374×
11,208×
12,075×
8,526×
3,257×
1,544×
28×

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×
314×
798×
587×
1,827×
5,805×
13,593×
13,327×
13,397×
12,233×
7,830×
11,562×
11,274×
9,490×
8,910×
9,830×
8,561×
6,318×
9,805×
10,882×
3,845×
246×
4,201×
13,778×
40,180×
24,556×
14,384×
12,365×
16,758×
13,741×
5,723×
4,950×
2,767×
1,684×
1,233×
1,789×
996×
230×

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:
0×
1×
782×
2,856×
6,541×
2,954×
664×
3,684×
6,608×
16,629×
23,143×
24,788×
16,253×
15,578×
25,468×
21,500×
9,374×
8,793×
9,614×
11,634×
18,197×
21,960×
16,699×
20,172×
21,319×
22,641×
15,064×
21,423×
21,059×
29,674×
24,524×
19,286×
11,580×
12,347×
17,387×
37,077×
45,396×
42,462×
27,207×
27,141×
28,989×
36,312×
24,898×
16,484×
7,567×
9,228×
8,014×
4,915×
1,144×
160×

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:
0×
1×
194×
300×
504×
1,204×
5,298×
4,860×
15,525×
82,181×
77,072×
64,929×
49,863×
51,870×
63,844×
80,187×
75,014×
67,074×
56,843×
57,252×
87,402×
73,620×
99,815×
80,422×
77,684×
74,805×
77,108×
83,341×
108,829×
122,873×
87,772×
71,720×
56,613×
43,263×
22,356×
12,130×
6,857×
4,580×
3,145×
7,696×
2,189×
1,427×
1,594×

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:
0×
1×
314×
540×
0×
2,772×
12,732×
17,339×
40,701×
65,335×
90,931×
157,806×
227,870×
258,521×
308,864×
357,551×
366,583×
346,766×
318,185×
233,105×
173,718×
149,532×
110,447×
127,482×
154,550×
157,777×
112,141×
106,497×
95,172×
114,355×
85,655×
67,877×
53,039×
24,259×
17,957×
21,312×
25,244×
22,689×
15,936×
8,553×
6,472×
820×

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:
0×
1×
14×
0×
215×
1,601×
9,051×
8,909×
3,823×
260×
0×
0×
0×
0×
0×
0×
0×
0×
0×
0×
7,254×
77,876×
522,740×
589,243×
527,844×
481,927×
654,723×
699,361×
758,869×
635,620×
505,828×
355,173×
349,615×
335,660×
257,178×
192,647×
216,104×
216,072×
195,014×
360,441×
361,939×
361,478×
246,823×
197,938×
151,679×
98,693×
83,216×
71,913×
67,358×
31,167×
19,475×
2,622×
336×

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:
0×
1×
314×
480×
0×
6,696×
49,956×
108,216×
148,804×
327,040×
322,037×
320,561×
290,306×
306,010×
402,779×
256,467×
210,420×
294,205×
380,830×
475,611×
392,834×
503,954×
574,005×
461,834×
356,423×
269,353×
381,814×
275,290×
143,730×
136,052×
192,979×
224,906×
227,925×
289,287×
212,347×
178,398×
152,345×
104,435×
156,898×
288,057×
272,832×
290,522×
310,489×
410,286×
400,604×
431,988×
454,873×
410,068×
445,142×
496,841×
445,401×
504,572×
494,607×
448,908×
361,111×
220,251×
181,523×
161,645×
176,134×
215,638×
205,592×
202,419×
219,153×
245,125×
246,306×
342,419×
365,450×
348,984×
383,350×
359,602×
304,413×
202,142×
52,342×
21,133×
2,836×

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:
0×
1×
314×
840×
215×
1,998×
14,680×
6,174×
56,670×
717,788×
1,441,707×
1,567,850×
1,424,983×
1,645,388×
1,637,013×
1,420,623×
1,129,448×
1,032,695×
1,111,674×
913,053×
933,907×
915,367×
1,483,373×
1,575,165×
1,479,538×
1,019,882×
772,877×
708,789×
1,089,622×
1,251,474×
1,284,626×
704,408×
1,044,311×
1,051,556×
1,266,372×
1,468,584×
1,480,441×
813,884×
567,143×
403,379×
423,306×
456,619×
445,818×
397,297×
200,051×
293,915×
451,128×
578,820×
383,082×
176,409×
141,137×
159,798×
88,166×
43,664×
36,249×
32,110×
39,020×
72,128×
52,940×
60,415×
43,368×
40,044×
27,324×
18,209×
8,691×
3,566×
5,354×
160×

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:
0×
1×
2,042×
21,036×
78,648×
94,995×
84,616×
129,901×
471,713×
1,784,993×
3,343,934×
3,992,542×
2,168,830×
3,242,626×
3,322,071×
2,961,740×
2,542,247×
2,579,548×
1,881,369×
2,091,431×
2,052,714×
1,659,484×
1,412,263×
1,699,775×
1,569,031×
1,344,950×
1,417,064×
915,418×
443,251×
1,305,448×
1,207,863×
3,209,951×
3,975,706×
4,257,158×
2,731,084×
2,066,890×
2,408,562×
4,001,389×
3,467,612×
3,705,432×
3,125,153×
2,511,819×
1,925,736×
2,361,048×
3,396,452×
4,107,694×
4,159,661×
3,641,273×
4,022,435×
3,575,283×
3,223,349×
1,932,052×
1,898,851×
1,751,344×
2,983,170×
4,596,699×
5,256,921×
3,340,488×
2,416,253×
1,707,622×
1,353,178×
1,574,138×
1,615,624×
1,508,318×
697,063×
360,531×
354,536×
264,723×
98,555×
11,784×
5,458×
136×

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:
0×
1×
314×
1,140×
0×
6,432×
120,632×
831,392×
2,136,083×
4,802,172×
7,377,606×
9,022,642×
9,888,565×
9,314,210×
8,926,873×
9,275,300×
9,380,952×
8,584,717×
7,714,982×
6,345,372×
6,548,476×
5,022,979×
5,182,769×
5,278,963×
5,166,463×
7,359,668×
10,119,580×
12,555,945×
13,816,626×
14,374,762×
11,091,426×
9,840,142×
9,534,623×
8,241,007×
6,474,994×
5,575,960×
5,117,635×
5,874,507×
4,820,953×
4,340,388×
3,957,555×
3,636,343×
3,067,766×
1,871,564×
1,420,397×
789,453×
230,639×
72,603×
41,366×
27,588×

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:
0×
1×
1,082×
22,380×
79,943×
21,597×
56,264×
22,286×
163,713×
3,119,164×
6,824,513×
8,119,872×
6,968,114×
8,880,427×
8,981,473×
9,883,237×
15,020,578×
12,374,535×
19,098,444×
20,984,641×
29,910,348×
25,349,771×
25,374,562×
20,499,378×
23,678,881×
23,054,909×
25,719,205×
20,787,064×
13,953,550×
12,326,272×
9,537,782×
13,266,705×
15,858,116×
15,401,973×
11,786,279×
10,625,217×
9,234,086×
12,884,826×
8,290,610×
6,232,644×
4,121,124×
3,447,716×
2,688,982×
1,778,687×
1,781,207×
1,708,202×
834,520×
392,074×
128,654×
140,559×
19,013×
420×

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:
0×
1×
314×
1,080×
0×
0×
14,256×
438,920×
2,535,536×
6,519,898×
11,295,577×
16,040,217×
23,976,161×
28,776,140×
29,034,318×
34,001,614×
34,099,239×
40,945,378×
46,595,218×
45,847,980×
50,507,541×
54,709,017×
57,989,906×
62,303,293×
62,680,428×
56,965,123×
43,112,890×
29,266,504×
20,527,036×
16,826,707×
12,697,741×
10,721,929×
6,553,559×
5,935,776×
4,688,838×
2,174,907×
833,918×
168,691×
22,724×
782×
43×

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:
0×
1×
494×
2,340×
695×
3,243×
41,269×
18,004×
208,371×
5,450,978×
12,321,640×
11,391,787×
5,656,919×
3,166,281×
1,779,824×
1,959,204×
2,582,582×
1,450,953×
909,631×
1,620,114×
1,872,441×
731,804×
478,817×
55,294×
1,576×
160×
0×
0×
52,005×
1,709,555×
21,004,750×
31,586,661×
43,422,445×
44,999,114×
29,931,962×
26,779,834×
18,907,665×
27,722,886×
29,083,170×
29,589,719×
30,764,467×
23,995,290×
16,411,041×
20,886,368×
28,662,398×
49,832,227×
45,282,868×
34,467,093×
24,741,190×
28,921,739×
24,790,557×
48,144,089×
56,730,553×
63,535,060×
34,313,164×
23,363,516×
15,926,869×
22,063,099×
25,421,358×
25,511,915×
15,420,894×
12,589,635×
8,892,823×
10,110,475×
10,419,535×
7,647,035×
6,110,088×
9,221,267×
14,109,469×
17,460,782×
14,414,846×
14,900,856×
16,600,033×
26,339,959×
28,137,481×
29,490,154×
28,082,724×
19,616,789×
14,718,855×
13,880,987×
16,144,467×
9,917,448×
9,732,038×
9,479,783×
9,459,197×
7,917,244×
5,607,929×
2,231,469×
1,237,219×
814,273×
430,688×
287,608×
191,851×
46,564×
19,328×
23,716×
11,082×

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:
0×
1×
314×
1,080×
0×
0×
0×
0×
0×
34,800×
1,243,648×
7,591,380×
29,013,680×
44,809,187×
82,739,022×
119,550,043×
148,114,514×
168,635,306×
167,246,921×
147,612,523×
131,237,970×
122,639,385×
116,109,660×
117,298,819×
118,239,176×
102,486,928×
104,965,496×
97,360,608×
93,634,567×
83,871,894×
76,020,166×
59,322,990×
36,136,863×
26,446,568×
20,014,231×
16,733,273×
18,666,624×
16,679,631×
15,665,987×
11,796,385×
6,681,856×
5,882,604×
3,022,826×
1,415,641×
673,657×
328,312×
32,004×
2,860×

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×
11,846×
754,824×
4,963,492×
22,951,288×
34,894,780×
44,678,547×
60,612,796×
133,676,101×
183,091,572×
261,235,273×
293,491,346×
269,779,329×
310,645,523×
388,283,948×
459,389,219×
545,337,213×
586,047,683×
539,074,907×
529,971,393×
406,948,365×
280,315,586×
203,499,065×
144,616,404×
113,726,782×
84,013,188×
75,490,863×
52,008,646×
35,171,504×
24,315,705×
11,429,681×
4,031,101×
1,931,541×
552,167×
89,059×
44,295×
11,655×
112×

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:
0×
1×
3,770×
221,160×
1,029,063×
4,347,053×
68,742,358×
151,611,759×
167,605,701×
221,965,714×
186,350,557×
171,273,635×
197,868,655×
276,322,276×
335,181,424×
284,725,011×
222,967,637×
201,702,574×
201,122,970×
246,424,899×
222,095,031×
217,705,665×
158,149,814×
121,486,788×
146,296,950×
182,846,107×
180,480,398×
175,112,843×
185,797,128×
142,853,334×
134,708,686×
116,531,846×
137,879,388×
176,868,350×
237,075,220×
140,152,996×
92,367,584×
76,845,431×
100,335,188×
81,325,229×
78,029,920×
67,417,773×
87,192,522×
104,964,203×
181,112,032×
150,843,343×
147,497,270×
135,609,603×
131,638,132×
147,545,044×
172,749,019×
206,657,017×
204,895,139×
262,788,741×
195,243,993×
148,194,935×
143,471,044×
181,852,662×
179,065,395×
129,715,501×
129,534,331×
88,194,818×
51,366,780×
62,191,087×
44,170,393×
38,733,269×
33,871,261×
28,461,777×
20,219,826×
31,983,164×
28,108,421×
15,359,127×
14,238,613×
16,459,096×
15,266,843×
13,142,486×
5,985,492×
3,259,663×
681,598×
717,640×
2,115,050×
1,870,960×
500,269×
154,337×
70,829×
27,001×
5,768×
720×