Base 13: Iteration Frequencies: Cycle Start

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
Cycle Start

0.92 step
avg

1.99 step
avg

3.38 step
avg

2.05 step
avg

6.65 step
avg

11.84 step
avg

5.99 step
avg

9.44 step
avg

12.98 step
avg

9.81 step
avg

14.75 step
avg

10.83 step
avg

19.65 step
avg

9.21 step
avg

22.09 step
avg

18.83 step
avg

14.23 step
avg

15.06 step
avg

21.67 step
avg

12.32 step
avg

11.83 step
avg

14.83 step
avg

15.79 step
avg

21.79 step
avg

11.67 step
avg

18.71 step
avg

12.27 step
avg

25.60 step
avg

11.92 step
avg

Step Frequency
Cycle Start

0s:
1s:
1×
12×

0s:
1s:
2s:
3s:
3×
31×
21×
36×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
1×
109×
45×
88×
78×
64×
46×
24×

0s:
1s:
2s:
3s:
4s:
1×
362×
1,043×
366×
48×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
1×
469×
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:
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:
1×
3,202×
4,528×
4,967×
4,894×
4,496×
5,192×
6,868×
5,733×
4,254×
2,306×
1,979×
1,370×
433×
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:
1×
581×
2,298×
7,604×
8,319×
8,606×
9,123×
9,803×
11,080×
9,778×
10,162×
9,798×
9,924×
8,643×
4,997×
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:
1×
4,993×
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:
1×
4,100×
15,287×
39,082×
47,476×
36,898×
56,143×
67,550×
81,580×
44,691×
33,165×
19,586×
12,096×
24,127×
25,276×
20,371×
16,875×
15,130×
20,627×
19,842×
15,766×
11,476×
6,356×
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:
1×
5,576×
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:
1×
3,908×
20,923×
83,047×
126,521×
134,900×
149,230×
193,391×
242,893×
232,015×
254,195×
211,011×
200,784×
152,254×
112,101×
96,645×
113,476×
90,504×
53,897×
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:
1×
13,548×
73,809×
134,302×
175,718×
212,224×
262,756×
249,691×
160,465×
141,030×
111,950×
93,993×
90,025×
107,780×
76,006×
88,528×
144,653×
141,975×
114,957×
85,723×
63,099×
49,234×
80,402×
110,640×
127,175×
198,524×
264,454×
272,238×
270,298×
200,878×
193,250×
155,446×
136,403×
130,789×
144,284×
90,405×
79,165×
36,377×
22,544×
37,448×
27,378×
14,307×
6,476×
3,348×
4,296×
2,308×

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×
7,704×
90,114×
356,515×
724,565×
681,069×
697,794×
885,648×
1,020,051×
970,272×
845,598×
693,762×
616,880×
782,523×
408,113×
257,978×
242,636×
147,359×
71,739×
71,603×
41,359×
16,134×
8,914×
8,236×
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:
1×
13,368×
108,406×
263,165×
397,842×
526,968×
469,433×
319,392×
323,857×
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:
1×
8,332×
117,395×
760,317×
1,962,004×
1,315,873×
1,041,577×
1,110,929×
1,339,887×
793,616×
695,183×
754,020×
715,267×
885,717×
1,482,887×
1,410,539×
1,656,764×
1,560,378×
1,177,427×
614,735×
790,590×
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:
1×
42,740×
515,944×
1,396,806×
2,268,624×
2,714,988×
2,792,894×
3,143,424×
3,244,117×
3,286,295×
3,426,687×
3,513,562×
2,885,802×
2,795,047×
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:
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:
1×
4,692×
97,880×
791,805×
2,971,745×
3,324,818×
2,384,381×
4,236,478×
5,797,173×
5,478,469×
3,370,969×
2,948,303×
2,647,145×
1,857,801×
1,931,847×
4,750,181×
4,844,869×
6,325,444×
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:
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:
1×
11,412×
249,233×
921,208×
2,085,504×
2,618,451×
4,394,777×
3,308,614×
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:
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:
1×
15,517×
414,127×
3,201,101×
13,395,148×
12,218,316×
11,019,266×
13,511,136×
16,265,629×
17,247,973×
10,459,489×
8,541,370×
12,732,483×
10,080,735×
9,707,257×
13,553,163×
11,324,017×
11,536,820×
12,269,029×
13,414,677×
8,629,651×
6,332,579×
3,957,251×
2,385,947×
1,592,225×
1,292,695×
423,233×
253,721×
18,170×
114×

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×
48,360×
1,542,212×
6,795,668×
21,108,646×
26,350,880×
25,609,323×
24,734,235×
24,591,475×
24,535,734×
24,192,112×
20,076,975×
22,551,213×
16,723,868×
14,731,613×
13,654,432×
8,472,482×
6,993,194×
6,019,669×
9,077,008×
9,536,500×
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×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
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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:
1×
16,164×
661,684×
5,582,471×
23,034,256×
25,276,415×
26,890,198×
28,503,787×
34,275,040×
38,280,432×
26,198,942×
33,805,812×
32,165,335×
21,213,241×
15,830,916×
20,282,427×
16,564,920×
26,453,342×
27,825,579×
22,269,701×
16,020,616×
11,834,623×
6,506,655×
4,929,802×
6,267,545×
3,390,208×
1,862,872×
889,951×
564,579×
3,534,543×
7,772,686×
15,841,995×
12,995,932×
9,149,272×
8,727,796×
4,858,401×
4,027,449×
2,571,644×
676,947×
249,366×
453,069×
94,772×
2,654×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
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16s:
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18s:
19s:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
1×
33,912×
1,265,004×
5,586,012×
18,512,224×
20,257,587×
15,678,719×
15,833,914×
20,360,309×
22,566,455×
33,978,928×
44,179,275×
51,321,020×
56,104,536×
52,162,160×
50,236,164×
42,262,911×
41,269,674×
41,828,350×
46,818,312×
47,225,853×
52,207,253×
44,634,175×
35,553,697×
27,795,734×
19,121,368×
12,019,398×
7,665,261×
3,941,133×
2,211,929×
1,189,223×
385,859×
201,476×
40,089×
3,885×

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1×
11,006×
375,866×
4,366,455×
31,056,196×
36,988,903×
28,340,467×
37,192,242×
45,688,527×
47,757,011×
34,030,326×
37,225,632×
36,629,478×
34,807,631×
35,894,537×
47,069,162×
39,234,822×
50,945,498×
53,964,219×
43,041,328×
41,072,374×
35,744,279×
21,663,290×
10,056,728×
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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1×
38,744×
2,351,408×
15,316,288×
70,613,240×
95,660,251×
113,436,783×
144,694,656×
143,982,390×
133,472,057×
122,793,998×
118,021,687×
97,490,269×
107,002,824×
113,941,043×
94,538,348×
124,586,209×
99,139,524×
91,536,990×
67,731,874×
40,582,255×
25,987,939×
11,097,246×
7,725,998×
6,758,837×
2,206,153×
820,252×
511,443×
267,184×
103,053×
61,904×
10,532×
1,616×

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1×
8,820×
450,231×
6,769,683×
53,544,383×
86,892,608×
63,975,256×
99,346,064×
132,670,432×
136,386,605×
97,122,611×
65,065,979×
81,194,822×
62,146,877×
54,892,696×
73,678,001×
114,323,481×
159,729,795×
155,422,256×
124,596,180×
88,557,271×
88,533,101×
87,377,101×
107,377,035×
95,362,677×
60,411,765×
49,620,642×
44,681,712×
45,355,194×
77,931,342×
55,443,603×
31,510,265×
42,786,407×
43,895,541×
43,393,360×
40,624,429×
37,491,267×
21,737,727×
11,337,471×
11,821,912×
6,800,194×
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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1×
21,944×
1,452,824×
9,717,972×
41,826,601×
86,446,565×
123,100,750×
184,678,586×
261,824,667×
291,879,265×
365,770,583×
380,714,528×
347,777,432×
377,673,168×
320,340,043×
289,207,772×
245,854,733×
189,993,209×
148,700,733×
94,767,679×
52,420,156×
34,745,060×
27,279,534×
15,972,954×
8,359,513×
5,651,879×
3,171,772×
991,936×
304,979×
136,114×
13,617×
867×

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1×
11,907×
776,681×
12,351,027×
103,910,934×
125,820,161×
80,725,852×
121,260,799×
139,361,458×
119,751,350×
74,718,561×
45,569,139×
43,970,131×
30,600,554×
30,012,091×
127,700,329×
139,621,593×
162,317,291×
140,966,557×
134,517,277×
161,153,921×
173,326,890×
156,035,001×
194,956,530×
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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1×
42,888×
4,218,764×
32,470,208×
166,589,426×
305,246,031×
361,405,680×
575,120,098×
642,956,652×
565,597,810×
715,951,311×
674,168,303×
633,188,652×
551,352,744×
510,541,722×
414,407,194×
314,573,555×
334,986,357×
287,553,969×
232,001,941×
202,316,158×
152,847,300×
86,600,926×
66,624,140×
31,722,378×
11,803,625×
6,772,844×
3,964,495×
6,085,630×
3,740,260×
2,582,556×
896,841×
298,354×
9,363×
14,823×
1,921×