Base 16: Iteration Frequencies: Cycle Start

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

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
Cycle Start

0.94 step
avg

1.38 step
avg

4.50 step
avg

3.26 step
avg

5.08 step
avg

11.92 step
avg

4.86 step
avg

11.81 step
avg

7.80 step
avg

10.22 step
avg

10.53 step
avg

9.53 step
avg

11.01 step
avg

10.12 step
avg

14.72 step
avg

10.84 step
avg

23.63 step
avg

11.70 step
avg

24.13 step
avg

12.61 step
avg

23.95 step
avg

13.51 step
avg

25.93 step
avg

14.20 step
avg

28.21 step
avg

15.01 step
avg

30.34 step
avg

Step Frequency
Cycle Start

0s:
1s:
1×
15×

0s:
1s:
2s:
5×
75×
56×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
1×
87×
70×
138×
130×
118×
102×
82×
58×
30×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
1×
486×
718×
1,031×
938×
348×
354×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
1×
1,607×
1,586×
1,052×
2,028×
1,656×
1,968×
2,986×
1,934×
656×
30×

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:
1×
1,780×
3,437×
4,470×
2,452×
2,666×
2,628×
2,842×
1,920×
2,876×
2,134×
2,631×
2,185×
1,912×
2,216×
1,655×
1,463×
995×
730×
1,500×
1,563×
910×
1,658×
878×
1,440×
811×
1,376×
1,474×
506×
551×
508×
96×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
1×
8,953×
17,104×
23,324×
23,538×
29,468×
26,884×
24,332×
11,734×
3,652×
1,004×
550×

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:
1×
5,727×
23,838×
29,450×
24,041×
20,053×
16,672×
24,821×
37,821×
41,938×
34,720×
31,057×
25,165×
18,695×
15,661×
17,536×
14,714×
14,563×
11,808×
7,522×
5,245×
5,586×
8,901×
7,550×
3,669×
4,608×
7,839×
4,605×
2,082×
2,993×
4,288×
7,754×
4,388×
2,468×
2,103×
432×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
1×
16,621×
55,444×
94,408×
113,272×
148,224×
164,796×
147,068×
93,172×
67,154×
62,548×
56,536×
71,838×
80,266×
48,808×
34,582×
30,216×
19,316×
3,106×
128×

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:
1×
22,820×
167,956×
245,089×
230,484×
194,991×
137,503×
167,074×
244,830×
228,312×
205,819×
233,027×
189,270×
151,249×
135,064×
113,204×
91,554×
90,591×
82,187×
65,947×
60,493×
39,261×
28,467×
32,743×
42,064×
27,569×
16,107×
10,923×
9,435×
2,981×
809×
936×

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:
1×
26,347×
139,714×
279,410×
328,960×
451,488×
591,944×
598,284×
542,234×
492,626×
589,158×
635,102×
507,360×
394,050×
427,474×
413,310×
273,648×
259,638×
252,534×
207,886×
153,794×
84,076×
43,346×
29,028×
4,748×

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:
1×
52,698×
649,871×
995,495×
1,017,930×
1,045,896×
1,094,923×
1,202,471×
1,209,749×
1,309,280×
1,450,123×
1,708,958×
1,404,825×
1,143,218×
881,097×
637,426×
445,661×
300,769×
269,144×
194,097×
147,008×
76,648×
59,105×
39,338×
30,033×
10,426×
5,516×
2,154×

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:
1×
23,559×
254,620×
763,618×
1,152,806×
1,788,018×
2,436,460×
2,773,394×
3,187,360×
3,386,596×
3,034,938×
2,537,684×
2,482,286×
2,581,404×
2,572,496×
2,129,878×
1,480,276×
1,208,286×
1,134,504×
858,074×
556,820×
507,460×
284,112×
194,598×
81,960×
29,048×
1,904×

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:
1×
115,646×
2,269,953×
3,468,342×
3,525,266×
4,032,651×
4,660,816×
5,708,961×
6,126,233×
6,719,066×
6,153,837×
6,660,295×
6,533,709×
5,545,151×
4,404,463×
2,851,816×
1,956,188×
1,317,672×
1,341,198×
1,130,216×
801,646×
558,011×
477,405×
285,993×
243,942×
148,428×
153,346×
112,025×
91,404×
53,460×
18,404×
27,831×
31,006×
11,152×
16,780×
4,237×
1,638×
572×

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:
1×
61,365×
1,094,058×
3,156,342×
5,107,814×
6,899,332×
8,181,078×
8,450,462×
7,922,594×
7,929,916×
8,559,088×
9,148,686×
8,400,192×
8,365,544×
7,724,234×
6,930,310×
6,701,624×
4,739,820×
3,575,512×
3,092,582×
2,611,006×
2,378,088×
2,184,254×
2,991,662×
2,716,406×
3,467,408×
2,893,126×
2,396,274×
2,384,870×
2,139,614×
2,535,832×
2,753,554×
2,640,684×
2,188,680×
1,349,662×
693,966×
468,454×
194,756×
74,040×
10,644×
3,986×

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:
1×
240,357×
7,673,666×
12,265,208×
9,734,176×
10,594,039×
14,982,175×
20,636,071×
23,476,834×
24,626,714×
24,911,093×
25,413,003×
22,433,485×
21,386,294×
18,725,003×
15,463,441×
11,357,741×
7,313,503×
7,258,392×
6,903,968×
4,889,904×
3,367,945×
2,811,554×
1,440,602×
735,536×
402,184×
488,735×
386,075×
350,266×
125,846×
57,331×
30,760×
44,074×
6,451×
5,126×
2,642×

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:
1×
33,327×
949,290×
3,819,340×
7,416,650×
13,005,908×
15,906,146×
18,838,532×
21,163,870×
18,681,096×
16,538,210×
13,302,820×
11,299,382×
12,332,702×
13,991,676×
15,183,684×
14,765,362×
11,429,012×
9,770,468×
8,204,584×
9,746,200×
11,612,418×
12,001,722×
12,861,224×
15,816,524×
17,925,500×
19,053,170×
16,880,732×
12,370,728×
10,970,242×
10,851,054×
12,651,200×
15,729,954×
14,921,378×
16,500,180×
18,695,848×
17,804,246×
16,256,644×
11,819,418×
10,276,020×
8,070,582×
5,988,814×
4,612,944×
4,201,886×
4,713,684×
3,892,338×
3,129,900×
4,347,436×
5,289,970×
4,208,232×
2,394,252×
1,159,176×
931,314×
988,556×
362,826×
52,908×
1,440×

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:
1×
493,581×
26,001,878×
45,258,339×
29,930,250×
27,089,227×
39,133,894×
56,614,044×
70,857,427×
78,960,930×
81,184,451×
85,828,070×
75,638,148×
69,751,160×
63,411,603×
57,772,058×
45,213,544×
31,779,348×
30,219,497×
26,227,936×
22,447,960×
19,876,132×
20,127,359×
10,578,748×
6,339,989×
3,960,694×
3,454,713×
2,790,027×
2,125,187×
1,014,859×
978,402×
661,607×
474,810×
166,450×
246,514×
206,737×
140,812×
18,980×
32,034×
53,036×
55,723×
12,747×
22,307×
5,991×
1,116×

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:
1×
100,359×
4,421,608×
18,162,682×
30,568,080×
44,120,716×
50,308,254×
62,696,098×
67,664,776×
58,987,284×
53,449,534×
46,882,440×
42,591,206×
44,751,176×
42,516,412×
44,923,452×
38,048,184×
32,294,762×
24,060,714×
27,123,056×
34,112,008×
33,698,276×
40,202,402×
45,641,066×
50,482,828×
48,020,996×
44,752,148×
44,465,334×
38,767,000×
42,266,150×
44,875,176×
52,155,162×
53,054,120×
58,602,244×
47,429,988×
43,627,712×
34,636,658×
34,096,726×
30,406,190×
29,524,838×
31,011,216×
31,444,396×
31,190,992×
26,234,116×
24,794,932×
22,498,950×
22,174,354×
24,381,382×
20,849,398×
16,924,762×
9,882,248×
4,396,764×
2,571,500×
1,555,894×
1,186,444×
359,646×
22,710×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
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17s:
18s:
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20s:
21s:
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23s:
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25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
43s:
1×
1,021,576×
89,785,081×
164,793,082×
98,223,161×
70,952,387×
94,284,710×
129,219,371×
161,536,038×
191,930,672×
213,532,013×
238,941,331×
224,796,660×
208,827,012×
201,013,730×
199,107,133×
164,307,410×
131,316,092×
121,105,776×
102,993,362×
89,005,636×
79,554,096×
78,014,355×
58,165,539×
37,747,613×
24,219,741×
22,537,793×
18,837,563×
12,373,425×
6,076,288×
5,654,289×
3,317,433×
1,608,618×
921,126×
1,127,693×
601,979×
248,365×
43,582×
85,832×
52,134×
46,456×
6,254×
7,046×
3,706×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
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45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
56s:
1×
274,995×
17,450,128×
74,900,764×
122,959,360×
160,605,174×
171,507,414×
186,999,692×
184,787,412×
155,027,708×
140,605,774×
139,293,552×
142,855,856×
130,700,682×
130,598,056×
117,602,346×
95,707,458×
69,357,498×
51,248,378×
52,932,636×
55,711,018×
74,121,826×
95,465,410×
119,246,654×
130,663,726×
144,855,324×
128,043,606×
116,340,058×
115,526,020×
128,225,946×
139,145,712×
169,917,414×
191,640,672×
213,761,934×
207,703,496×
212,464,262×
178,394,534×
168,375,962×
173,454,956×
149,137,514×
110,537,510×
88,485,168×
60,505,292×
49,323,074×
40,525,906×
29,028,252×
24,960,812×
27,106,762×
34,361,586×
25,318,180×
10,958,868×
5,534,700×
2,033,574×
958,972×
512,294×
137,622×
3,060×

0s:
1s:
2s:
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46s:
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48s:
49s:
50s:
51s:
52s:
1×
2,101,092×
297,502,623×
580,321,859×
331,879,071×
186,677,322×
203,631,172×
270,927,764×
334,835,762×
402,922,476×
470,673,287×
562,918,475×
566,585,657×
567,086,570×
571,500,842×
567,174,970×
499,860,641×
426,248,008×
411,297,523×
355,339,261×
289,142,978×
245,227,232×
247,347,462×
220,891,002×
186,365,058×
135,739,353×
113,342,224×
102,451,114×
68,024,888×
39,870,168×
33,235,585×
20,479,223×
17,296,996×
9,841,953×
8,069,283×
6,078,005×
3,845,046×
1,994,482×
2,015,664×
1,229,377×
753,428×
199,213×
325,950×
393,359×
217,160×
21,754×
64,600×
100,646×
81,644×
15,836×
44,210×
8,991×
1,500×

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1×
663,447×
60,413,568×
270,420,018×
402,263,466×
474,970,852×
492,323,676×
531,696,396×
527,369,422×
482,246,566×
458,078,842×
419,149,180×
403,414,472×
389,289,354×
378,338,868×
350,410,572×
304,482,256×
292,874,898×
242,616,128×
172,563,496×
152,266,864×
156,974,764×
189,506,840×
198,780,106×
194,520,428×
201,673,210×
204,617,192×
180,333,710×
154,922,684×
167,288,094×
188,103,054×
207,400,832×
206,522,790×
247,850,236×
297,296,832×
334,240,818×
331,946,500×
354,411,164×
419,648,784×
556,928,700×
518,221,760×
463,370,650×
384,915,498×
343,093,162×
288,761,544×
275,095,638×
226,402,650×
189,093,614×
152,323,064×
130,625,950×
131,287,824×
100,064,864×
75,393,446×
58,752,516×
66,996,060×
64,505,462×
73,087,116×
54,580,126×
65,333,596×
84,598,526×
68,147,168×
33,416,966×
14,849,964×
7,827,490×
1,566,906×
155,712×
208×

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1×
4,093,407×
919,805,928×
1,829,366,319×
1,123,446,185×
548,422,976×
479,935,292×
567,295,087×
625,074,044×
771,458,849×
904,441,468×
1,139,130,579×
1,220,775,235×
1,300,009,664×
1,334,105,388×
1,423,839,252×
1,336,004,457×
1,212,917,051×
1,228,979,798×
1,178,060,586×
1,003,475,325×
848,578,197×
768,812,174×
684,987,623×
542,325,229×
432,915,600×
387,252,892×
334,031,352×
233,394,163×
161,486,559×
161,209,215×
126,908,305×
105,508,231×
58,215,691×
53,918,590×
38,895,900×
22,237,105×
8,741,190×
8,602,975×
4,812,297×
2,695,411×
1,251,136×
1,735,407×
876,381×
430,355×
64,601×
151,423×
79,164×
65,770×
7,097×
8,966×
4,770×

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1×
1,353,975×
175,067,902×
814,288,948×
1,133,514,890×
1,215,370,318×
1,133,213,634×
1,195,015,674×
1,083,945,154×
959,850,186×
900,480,900×
949,104,242×
1,056,684,516×
1,153,269,642×
1,217,904,978×
1,197,582,222×
1,129,369,520×
1,024,388,614×
996,758,794×
954,693,118×
879,509,310×
744,002,244×
654,892,836×
535,989,400×
385,858,706×
314,878,052×
303,999,962×
323,100,262×
313,535,534×
344,797,898×
315,041,928×
292,213,514×
215,817,938×
220,842,932×
212,325,446×
265,904,122×
312,960,722×
395,755,880×
475,586,618×
596,240,760×
678,306,062×
769,217,572×
713,418,044×
774,418,546×
676,185,574×
643,292,790×
550,805,120×
557,091,710×
432,314,910×
479,704,822×
522,705,272×
583,823,340×
567,666,258×
583,320,420×
554,577,502×
581,123,394×
690,619,736×
580,108,160×
482,273,880×
383,434,760×
317,978,390×
280,693,300×
279,625,914×
181,076,866×
152,965,452×
221,731,252×
207,889,588×
132,064,836×
76,534,592×
41,747,322×
38,246,682×
39,254,398×
24,748,548×
14,352,028×
12,902,250×
3,269,202×
571,288×
157,934×
15,572×
478×

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1×
7,707,862×
2,633,924,370×
5,290,466,400×
3,375,924,685×
1,567,528,550×
1,098,012,249×
1,156,603,580×
1,252,681,118×
1,473,963,401×
1,691,563,845×
2,186,005,085×
2,283,944,703×
2,576,218,088×
2,729,115,715×
3,071,520,631×
3,061,206,663×
2,859,413,015×
2,889,157,236×
2,930,214,086×
2,642,393,289×
2,451,764,790×
2,351,400,425×
1,960,468,458×
1,495,322,894×
1,362,422,214×
1,318,449,542×
1,198,274,628×
891,747,891×
597,850,211×
556,586,523×
507,961,214×
425,887,499×
330,625,729×
371,802,703×
315,695,175×
198,041,776×
119,970,737×
67,595,252×
39,242,963×
31,601,944×
16,842,981×
15,680,384×
9,984,821×
6,668,638×
2,903,179×
3,815,366×
2,242,689×
1,337,707×
265,336×
593,999×
688,788×
397,142×
30,436×
113,550×
170,080×
151,691×
19,357×
77,225×
12,503×
1,884×

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1×
2,443,711×
436,274,044×
2,031,275,116×
2,584,727,880×
2,489,654,202×
2,330,599,476×
2,547,154,114×
2,390,441,034×
2,006,200,168×
1,740,255,140×
1,602,279,186×
1,672,237,068×
2,051,101,530×
2,494,803,788×
2,906,205,358×
2,665,877,546×
2,560,747,928×
2,840,311,680×
3,017,164,038×
3,135,652,038×
2,951,217,154×
2,502,671,578×
1,821,630,748×
1,443,712,560×
1,196,406,800×
1,127,897,314×
1,058,846,872×
1,052,164,148×
1,095,195,570×
982,041,120×
1,047,071,570×
887,696,342×
799,622,826×
720,101,958×
646,567,670×
646,945,480×
698,920,532×
723,754,370×
782,243,142×
921,439,882×
1,190,267,024×
1,278,110,126×
1,570,875,290×
1,422,643,426×
1,357,801,322×
980,687,316×
864,462,610×
563,055,528×
464,203,906×
422,125,702×
447,756,750×
379,152,010×
446,013,616×
370,024,942×
530,311,080×
631,360,850×
801,190,906×
957,942,152×
1,139,564,000×
1,225,736,582×
1,369,564,578×
1,857,761,954×
1,888,093,072×
1,691,547,806×
1,543,416,300×
1,341,799,144×
1,166,813,742×
1,190,145,044×
946,970,488×
660,781,734×
519,429,750×
336,633,994×
182,050,346×
138,175,636×
82,294,768×
43,617,878×
36,561,214×
16,541,182×
3,218,206×
1,852,018×
311,866×
9,076×