Base 12: Iteration Frequencies: Cycle Start

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

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

1.33 step
avg

3.69 step
avg

3.07 step
avg

6.65 step
avg

7.93 step
avg

10.89 step
avg

8.12 step
avg

6.60 step
avg

12.71 step
avg

8.66 step
avg

7.95 step
avg

8.15 step
avg

7.99 step
avg

12.15 step
avg

6.89 step
avg

14.09 step
avg

6.21 step
avg

14.16 step
avg

6.21 step
avg

12.91 step
avg

6.06 step
avg

13.76 step
avg

7.64 step
avg

14.26 step
avg

7.07 step
avg

14.82 step
avg

6.57 step
avg

14.99 step
avg

7.39 step
avg

14.50 step
avg

7.38 step
avg

13.51 step
avg

Step Frequency
Cycle Start

0s:
1s:
1×
11×

0s:
1s:
2s:
4×
44×
30×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
1×
53×
40×
78×
70×
58×
42×
22×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
1×
244×
281×
364×
232×
125×
82×
36×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
1×
223×
202×
286×
422×
502×
382×
426×
420×
594×
658×
230×
22×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
1×
553×
920×
1,314×
823×
1,139×
728×
857×
1,134×
770×
501×
399×
530×
651×
392×
599×
525×
396×
124×
20×

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×
1,497×
2,272×
3,122×
2,468×
1,230×
318×
522×
454×
888×
586×
1,122×
1,004×
1,232×
1,970×
3,146×
4,042×
2,560×
824×
376×
508×
754×
672×
186×
70×

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×
2,207×
7,016×
8,731×
7,168×
5,394×
3,729×
4,773×
4,184×
4,492×
3,795×
3,064×
3,459×
4,709×
3,197×
2,866×
2,421×
1,199×
435×
576×
846×
388×
444×
454×
34×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
1×
5,499×
13,826×
19,640×
19,782×
21,252×
21,822×
15,918×
9,362×
7,334×
4,694×
3,762×
5,552×
4,010×
3,884×
4,944×
2,880×
2,804×
994×

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:
1×
1,781×
12,502×
27,137×
23,593×
23,671×
24,069×
19,304×
23,195×
16,533×
12,840×
12,887×
10,130×
9,128×
11,561×
8,573×
10,404×
7,433×
8,046×
8,572×
8,248×
4,742×
4,261×
5,413×
4,645×
4,266×
5,160×
9,034×
10,445×
8,200×
5,062×
3,699×
2,320×
2,609×
1,960×
972×
320×

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:
1×
8,399×
35,232×
47,714×
57,566×
55,716×
59,654×
63,390×
59,752×
66,594×
50,056×
35,878×
28,410×
30,026×
16,250×
14,354×
20,522×
16,108×
10,786×
8,130×
2,308×
1,490×
2,718×
5,996×
7,078×
1,304×

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×
5,504×
74,704×
174,491×
172,254×
144,189×
117,155×
86,853×
72,496×
59,184×
65,491×
63,307×
55,439×
50,604×
39,831×
36,645×
37,320×
26,693×
20,342×
13,344×
11,313×
10,949×
6,843×
1,813×
969×
1,448×
2,120×
776×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
1×
17,879×
139,272×
208,298×
212,918×
197,434×
186,086×
203,544×
221,500×
228,254×
202,914×
122,306×
141,756×
102,844×
107,494×
83,300×
65,324×
29,170×
8,526×
4,596×
11,294×
1,434×

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×
9,909×
240,348×
613,161×
595,445×
444,248×
399,111×
349,548×
289,671×
231,592×
194,882×
147,937×
129,878×
114,790×
96,223×
82,317×
83,090×
77,079×
84,543×
68,918×
53,351×
27,402×
26,231×
23,566×
16,803×
15,831×
9,118×
9,772×
5,729×
4,109×
2,400×
3,984×
2,896×
2,969×
548×

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:
1×
6,177×
84,420×
249,048×
303,820×
370,254×
462,736×
503,106×
514,250×
567,016×
497,496×
386,088×
326,802×
348,776×
350,800×
418,318×
353,410×
354,660×
343,752×
354,118×
272,204×
175,572×
114,126×
83,206×
74,264×
84,660×
56,862×
25,896×
16,234×
8,224×
6,914×
12,196×
754×

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×
22,114×
868,050×
2,094,499×
1,959,504×
1,468,693×
1,176,944×
1,029,126×
856,418×
650,519×
568,051×
441,651×
391,659×
323,889×
270,814×
210,077×
177,681×
158,791×
110,407×
86,036×
41,101×
28,116×
20,241×
13,228×
13,926×
11,735×
7,873×
12,576×
7,657×
3,985×
2,904×
5,992×
2,224×
1,413×

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:
1×
14,587×
366,440×
1,088,856×
1,126,820×
1,072,658×
1,275,062×
1,168,206×
961,682×
869,006×
583,322×
621,852×
506,308×
596,672×
624,076×
763,792×
802,780×
784,548×
874,410×
1,065,942×
1,063,862×
1,047,260×
982,962×
706,028×
596,524×
468,080×
397,926×
255,594×
193,086×
120,732×
113,854×
109,042×
115,894×
62,798×
40,510×
23,164×
6,020×
3,548×
276×

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×
47,265×
2,743,324×
6,294,559×
5,561,435×
4,272,863×
3,359,713×
2,678,815×
2,320,951×
1,576,027×
1,246,329×
901,976×
875,829×
692,292×
614,662×
377,295×
327,777×
243,210×
157,588×
96,380×
51,091×
38,867×
28,087×
18,133×
14,444×
10,230×
7,661×
11,781×
8,537×
5,225×
3,408×
6,400×
3,280×
1,855×

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×
14,523×
505,620×
1,575,466×
1,682,600×
2,070,756×
2,589,280×
2,819,492×
2,775,040×
2,553,832×
2,393,566×
2,517,750×
2,283,062×
2,279,848×
2,126,782×
2,431,046×
2,727,866×
2,589,580×
2,714,508×
2,749,056×
2,488,360×
2,346,224×
1,886,840×
1,515,122×
1,680,384×
997,230×
695,030×
466,290×
290,864×
236,794×
174,462×
141,424×
104,586×
60,496×
50,030×
29,732×
20,804×
11,144×
11,852×
11,080×
8,878×

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×
89,497×
7,590,934×
16,168,222×
14,037,480×
9,579,456×
7,210,807×
5,703,299×
5,238,829×
4,289,377×
3,100,115×
2,205,911×
2,175,336×
1,833,680×
1,535,159×
1,186,192×
880,001×
622,637×
395,572×
255,129×
119,835×
93,839×
81,517×
54,919×
30,700×
29,032×
38,067×
54,625×
22,480×
28,043×
5,208×
9,886×
4,295×
2,235×

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×
42,533×
2,218,342×
7,143,960×
6,980,496×
7,602,952×
8,113,800×
7,153,762×
6,495,178×
5,568,538×
5,479,522×
5,856,044×
5,780,154×
4,803,022×
4,560,712×
4,711,256×
6,429,382×
5,686,770×
4,710,250×
4,571,794×
4,106,118×
3,799,198×
3,269,934×
2,514,648×
1,980,404×
1,430,638×
1,190,010×
1,024,510×
989,908×
1,111,538×
805,686×
635,314×
405,994×
366,036×
430,376×
389,132×
315,480×
124,634×
66,146×
59,948×
35,602×
18,412×
12,586×
12,072×
13,122×
8,566×

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×
158,773×
19,171,886×
38,816,284×
32,512,879×
21,644,346×
15,892,049×
12,454,329×
10,867,334×
9,583,401×
7,269,590×
5,411,474×
5,005,747×
4,303,613×
2,800,656×
1,808,313×
1,268,381×
1,252,993×
929,500×
575,993×
446,211×
305,142×
368,777×
232,103×
105,430×
76,588×
71,666×
73,706×
43,717×
19,205×
17,248×
24,808×
15,565×
4,603×
1,070×
1,182×
1,222×
935×

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:
1×
52,891×
3,728,278×
12,663,900×
12,400,400×
14,748,758×
15,786,268×
13,853,464×
14,263,806×
12,043,398×
10,994,090×
12,975,192×
12,451,060×
10,006,096×
10,799,240×
11,267,488×
14,974,388×
14,184,648×
12,421,950×
10,529,972×
9,351,308×
10,252,892×
8,837,830×
7,017,320×
5,898,760×
5,271,046×
4,542,512×
3,181,862×
2,290,110×
2,380,010×
1,638,664×
1,431,238×
833,808×
708,326×
600,114×
525,232×
299,084×
202,308×
130,356×
107,154×
96,540×
63,042×
67,776×
54,678×
66,988×
34,354×
15,832×
13,956×
15,044×
14,508×
9,820×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
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29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
42s:
1×
260,727×
44,049,780×
84,684,444×
67,309,014×
40,135,199×
27,077,458×
21,189,785×
18,444,933×
15,022,217×
12,900,779×
9,947,098×
9,122,945×
5,525,699×
2,880,212×
2,382,733×
2,448,125×
2,626,384×
3,084,525×
3,107,598×
4,105,716×
4,136,107×
5,041,621×
5,638,655×
5,722,082×
5,998,586×
6,021,308×
4,419,969×
2,377,520×
794,020×
290,317×
166,684×
124,457×
38,347×
22,662×
34,140×
57,490×
29,510×
2,644×
1,070×
1,182×
1,222×
935×

0s:
1s:
2s:
3s:
4s:
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1×
82,015×
7,731,504×
28,446,688×
25,059,078×
28,716,356×
31,189,888×
26,812,000×
28,940,172×
25,278,916×
23,754,140×
25,120,346×
23,037,970×
19,174,334×
19,983,164×
22,770,428×
31,520,632×
32,231,860×
29,553,330×
23,601,902×
18,825,546×
17,916,344×
15,555,730×
12,822,472×
12,807,750×
10,625,278×
12,693,094×
12,494,662×
8,515,968×
6,195,176×
4,517,524×
3,664,360×
2,663,316×
1,695,756×
1,369,466×
1,004,048×
842,434×
543,754×
546,836×
635,276×
613,330×
275,762×
207,048×
130,006×
121,458×
108,858×
87,836×
75,206×
63,496×
70,066×
40,326×
16,676×
15,572×
14,904×
16,036×
9,202×

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1×
417,295×
95,363,814×
177,045,604×
139,876,101×
84,154,869×
59,058,313×
43,326,476×
31,897,318×
24,892,851×
19,780,695×
14,088,708×
15,256,917×
14,488,832×
14,000,426×
15,237,971×
16,929,885×
18,786,792×
15,785,585×
11,144,913×
9,185,078×
7,700,881×
9,934,720×
8,442,682×
3,352,356×
1,451,787×
809,172×
723,558×
454,060×
220,784×
207,594×
244,270×
260,418×
137,735×
62,366×
57,914×
62,387×
59,076×
32,372×
13,166×
12,790×
15,992×
11,283×
1,936×
1,070×
1,182×
1,222×
935×

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1×
131,429×
15,501,882×
61,860,338×
56,189,194×
57,607,892×
60,862,480×
55,047,144×
59,321,414×
60,162,572×
55,069,360×
51,501,842×
45,577,652×
36,954,780×
32,728,068×
37,851,916×
51,216,140×
53,760,806×
44,303,894×
33,186,668×
27,276,236×
29,086,044×
26,235,496×
23,796,642×
26,224,294×
22,473,272×
26,015,764×
27,662,286×
22,338,202×
17,731,858×
11,846,848×
10,599,978×
8,609,392×
6,356,602×
6,636,166×
6,297,688×
8,631,288×
9,359,940×
5,374,174×
4,142,878×
2,199,480×
1,298,538×
850,572×
752,748×
660,300×
672,642×
304,790×
227,078×
132,242×
131,138×
118,732×
86,172×
84,292×
66,182×
83,932×
41,086×
18,198×
17,188×
18,236×
17,544×
10,678×

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1×
647,264×
196,390,834×
356,827,493×
274,596,162×
164,417,580×
118,892,613×
95,373,965×
69,450,420×
56,020,166×
42,197,700×
33,407,173×
33,323,806×
31,901,948×
31,208,346×
30,947,224×
28,018,252×
27,550,541×
22,669,825×
17,653,599×
11,982,714×
9,876,245×
8,007,468×
3,267,081×
1,326,747×
1,352,156×
1,410,721×
1,588,802×
1,313,524×
695,620×
630,829×
632,291×
677,246×
430,591×
211,010×
202,170×
233,038×
254,984×
134,204×
62,366×
57,914×
62,387×
59,076×
32,372×
13,166×
12,790×
15,992×
11,283×
1,936×
1,070×
1,182×
1,222×
935×

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1×
221,787×
33,836,890×
142,039,900×
133,892,402×
123,085,808×
132,797,128×
121,474,584×
133,904,808×
122,680,250×
113,361,968×
96,139,308×
75,059,280×
52,987,556×
48,557,654×
57,074,502×
76,678,562×
78,795,658×
65,293,224×
52,323,332×
41,258,838×
42,604,606×
36,679,602×
35,171,776×
32,997,468×
33,134,404×
45,742,282×
53,692,480×
42,640,372×
33,462,740×
23,931,742×
22,766,382×
16,245,140×
14,440,192×
17,484,160×
16,387,902×
21,062,926×
25,710,624×
19,077,004×
14,107,356×
9,173,402×
7,112,800×
4,748,284×
4,620,322×
4,720,040×
5,103,688×
7,719,748×
7,752,090×
5,518,562×
2,917,512×
1,741,878×
593,606×
365,440×
171,668×
146,920×
131,564×
70,248×
93,228×
76,976×
86,822×
47,970×
20,032×
18,804×
17,736×
19,580×
9,922×

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1×
945,511×
378,568,912×
644,656,003×
471,485,401×
269,766,308×
199,236,557×
160,134,281×
123,761,997×
104,873,250×
85,617,192×
72,693,078×
73,296,799×
63,465,285×
59,314,255×
58,334,097×
54,832,437×
55,566,369×
47,353,589×
36,428,171×
30,825,816×
25,058,795×
23,545,357×
22,486,868×
29,231,888×
19,921,562×
14,871,974×
11,429,718×
6,372,029×
3,540,195×
2,585,165×
1,747,084×
1,672,786×
1,429,990×
847,873×
516,797×
589,175×
664,932×
425,374×
211,010×
202,170×
233,038×
254,984×
134,204×
62,366×
57,914×
62,387×
59,076×
32,372×
13,166×
12,790×
15,992×
11,283×
1,936×
1,070×
1,182×
1,222×
935×

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1×
385,051×
74,753,170×
331,854,336×
297,881,584×
246,954,504×
260,853,794×
243,789,294×
283,549,880×
274,968,142×
256,959,262×
198,112,010×
131,403,056×
83,602,990×
74,238,826×
77,725,144×
106,879,080×
101,777,802×
78,517,788×
66,460,412×
54,732,052×
59,706,606×
51,144,106×
45,446,602×
43,983,068×
48,668,206×
65,181,118×
68,178,808×
58,429,498×
46,834,192×
38,279,526×
38,054,844×
33,149,312×
30,499,016×
33,030,378×
33,728,738×
40,019,870×
38,466,774×
32,546,902×
27,285,060×
19,376,392×
16,417,426×
10,516,368×
9,939,334×
13,987,736×
13,869,228×
18,583,688×
20,510,634×
15,125,714×
9,871,598×
8,283,104×
5,654,392×
2,708,472×
3,929,216×
3,931,444×
4,576,100×
7,124,596×
7,099,084×
4,713,148×
3,206,864×
1,161,808×
463,360×
709,696×
373,700×
189,540×
103,384×
29,702×
20,994×
21,428×
20,846×
11,578×

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1×
1,379,636×
710,087,438×
1,198,840,833×
875,093,550×
493,426,317×
371,954,698×
271,868,268×
218,039,323×
176,607,845×
148,922,743×
129,415,692×
135,567,318×
119,753,853×
106,790,446×
93,952,356×
110,801,590×
99,951,165×
83,816,665×
64,559,210×
51,663,095×
52,144,190×
42,886,598×
26,151,938×
16,840,822×
12,138,455×
11,313,691×
18,025,531×
16,228,183×
14,867,903×
9,781,854×
7,857,302×
7,827,659×
6,526,831×
9,211,306×
10,822,188×
9,483,598×
8,254,306×
3,671,831×
1,470,767×
878,192×
669,675×
665,016×
425,374×
211,010×
202,170×
233,038×
254,984×
134,204×
62,366×
57,914×
62,387×
59,076×
32,372×
13,166×
12,790×
15,992×
11,283×
1,936×
1,070×
1,182×
1,222×
935×

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1×
664,203×
162,410,752×
710,524,454×
645,964,786×
514,995,552×
529,089,188×
515,115,070×
560,495,982×
501,086,596×
469,258,320×
362,707,354×
211,796,954×
132,312,544×
127,920,144×
140,588,870×
151,895,532×
136,558,978×
100,739,410×
79,795,278×
69,926,120×
75,413,006×
69,341,786×
58,392,986×
55,729,634×
65,628,310×
95,765,436×
91,958,300×
68,585,202×
61,952,432×
52,938,448×
49,902,354×
39,062,064×
35,709,352×
39,834,078×
42,363,372×
58,247,818×
60,403,824×
47,158,480×
40,993,862×
29,338,194×
27,168,336×
19,314,418×
20,765,984×
23,874,262×
22,529,442×
34,264,412×
32,690,146×
27,187,330×
24,819,932×
16,046,272×
13,163,060×
7,237,794×
9,356,688×
10,569,138×
12,039,394×
17,467,408×
16,646,766×
11,723,030×
12,804,800×
8,501,370×
4,248,390×
2,287,702×
2,963,508×
3,682,154×
4,703,036×
7,134,250×
6,085,028×
4,316,396×
1,955,366×
833,636×
168,614×
153,990×
37,740×
23,672×
10,642×