Base 16: Iteration Frequencies: Full Cycle

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
Full Cycle

1.94 step
avg

5.02 step
avg

5.50 step
avg

7.87 step
avg

8.37 step
avg

16.83 step
avg

14.48 step
avg

17.42 step
avg

14.94 step
avg

16.29 step
avg

17.32 step
avg

15.52 step
avg

14.37 step
avg

15.92 step
avg

28.12 step
avg

16.52 step
avg

24.73 step
avg

17.20 step
avg

26.21 step
avg

18.04 step
avg

26.43 step
avg

18.87 step
avg

28.96 step
avg

19.50 step
avg

31.56 step
avg

20.24 step
avg

34.14 step
avg

Step Frequency
Full Cycle

0s:
1s:
2s:
0×
1×
15×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
0×
1×
15×
0×
4×
60×
56×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
0×
1×
87×
70×
138×
130×
118×
102×
82×
58×
30×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
0×
1×
15×
63×
215×
274×
453×
691×
464×
728×
570×
278×
124×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
0×
1×
15×
198×
654×
1,404×
1,242×
1,420×
2,108×
2,718×
3,124×
1,934×
656×
30×

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:
0×
1×
87×
81×
145×
568×
2,442×
2,992×
3,718×
2,332×
2,880×
2,224×
2,834×
1,964×
2,812×
2,126×
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:
12s:
13s:
14s:
15s:
16s:
17s:
18s:
19s:
20s:
21s:
22s:
0×
1×
515×
1,104×
2,542×
1,776×
3,506×
1,308×
1,372×
1,158×
3,754×
7,108×
14,382×
16,696×
20,436×
20,322×
22,194×
20,782×
17,640×
9,006×
3,388×
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:
36s:
37s:
38s:
39s:
0×
1×
15×
81×
193×
3,875×
16,501×
16,492×
15,529×
12,307×
14,716×
22,909×
20,654×
24,713×
27,411×
31,398×
38,342×
35,914×
22,914×
24,048×
23,649×
20,126×
19,160×
14,160×
8,438×
9,531×
10,705×
8,929×
4,043×
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:
20s:
21s:
22s:
23s:
24s:
25s:
26s:
0×
1×
1,047×
4,482×
8,690×
3,586×
1,280×
7,372×
22,426×
44,896×
79,440×
107,752×
154,880×
158,432×
108,066×
93,918×
71,020×
64,498×
57,876×
62,014×
62,012×
71,242×
44,886×
40,172×
29,904×
7,484×
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:
32s:
33s:
34s:
35s:
36s:
37s:
0×
1×
330×
2,101×
865×
12,705×
97,413×
135,575×
159,056×
148,300×
149,664×
205,955×
203,966×
177,617×
135,873×
149,616×
191,901×
190,921×
189,607×
183,162×
128,702×
104,543×
99,081×
105,105×
83,249×
68,470×
62,259×
52,321×
56,587×
39,782×
24,624×
28,030×
36,694×
20,373×
12,526×
6,199×
3,967×
1,620×

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:
0×
1×
1,743×
9,780×
26,362×
36,236×
87,220×
179,212×
280,888×
258,752×
269,284×
373,154×
390,570×
453,810×
403,178×
366,360×
392,970×
364,386×
441,468×
449,544×
463,030×
362,184×
390,834×
340,160×
320,398×
246,080×
220,894×
252,220×
181,482×
85,142×
33,944×
23,364×
18,946×
2,564×

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:
0×
1×
1,077×
11,531×
40,493×
66,534×
384,436×
503,455×
663,054×
727,779×
948,133×
1,165,924×
1,027,851×
1,039,197×
1,059,500×
1,206,148×
1,056,949×
1,045,206×
1,163,304×
1,214,653×
1,039,578×
794,492×
584,503×
463,085×
329,204×
227,866×
195,018×
141,383×
111,190×
61,541×
52,980×
28,855×
18,402×
7,652×
2,310×
576×

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:
0×
1×
1,743×
18,108×
95,672×
221,958×
501,954×
965,804×
1,479,530×
2,179,080×
2,870,232×
3,214,982×
3,258,848×
3,144,196×
3,111,468×
2,692,762×
2,191,876×
2,107,136×
2,033,692×
1,614,782×
1,201,272×
1,081,046×
1,040,350×
791,382×
535,462×
497,328×
283,986×
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:
38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
0×
1×
4,675×
73,729×
351,281×
319,009×
1,459,913×
1,937,761×
2,178,932×
2,727,537×
3,795,504×
4,837,200×
4,999,885×
5,292,606×
4,689,745×
5,324,907×
5,551,189×
5,616,100×
5,147,968×
4,498,097×
4,295,111×
3,274,612×
2,713,614×
1,894,669×
1,234,595×
871,010×
1,003,198×
911,996×
621,807×
429,077×
415,744×
261,115×
221,148×
122,239×
137,870×
101,636×
80,620×
51,040×
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:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
0×
1×
6,189×
138,810×
617,848×
1,271,314×
2,234,072×
2,449,992×
2,461,562×
2,333,418×
2,331,252×
1,740,884×
1,150,972×
697,820×
631,664×
385,164×
183,060×
914,202×
2,164,720×
3,429,972×
4,333,020×
5,511,178×
5,825,158×
5,233,740×
5,980,334×
7,413,068×
8,422,732×
7,765,682×
7,980,334×
7,587,734×
6,811,812×
6,632,076×
4,733,226×
3,574,962×
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:
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16s:
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18s:
19s:
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21s:
22s:
23s:
24s:
25s:
26s:
27s:
28s:
29s:
30s:
31s:
32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:
40s:
41s:
0×
1×
9,213×
240,269×
1,384,359×
1,773,795×
5,462,425×
7,610,739×
6,789,593×
7,625,062×
11,982,192×
16,885,268×
17,352,076×
17,662,282×
19,049,433×
21,950,013×
21,164,618×
22,069,163×
20,059,802×
16,787,291×
15,533,462×
12,633,510×
11,389,183×
9,866,587×
6,705,674×
4,468,735×
5,506,937×
5,788,832×
3,977,262×
2,855,591×
2,475,670×
1,237,360×
602,617×
338,149×
446,163×
336,122×
317,566×
121,781×
49,448×
14,614×
14,701×
2,637×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
8s:
9s:
10s:
11s:
12s:
13s:
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24s:
25s:
26s:
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31s:
32s:
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35s:
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37s:
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39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
56s:
57s:
0×
1×
7,575×
221,898×
1,774,626×
4,707,230×
9,387,006×
15,305,540×
18,120,564×
23,200,720×
22,109,060×
18,984,894×
14,546,684×
12,095,548×
12,711,394×
14,066,034×
15,222,442×
14,766,704×
11,429,088×
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:
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34s:
35s:
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39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
50s:
51s:
52s:
0×
1×
11,181×
569,229×
4,466,901×
7,266,597×
21,923,085×
33,481,658×
21,365,138×
20,440,173×
32,121,377×
47,248,858×
53,856,748×
59,234,778×
62,879,255×
71,718,728×
72,224,721×
72,192,732×
66,726,540×
60,050,412×
51,142,033×
40,548,603×
37,650,214×
33,698,559×
25,280,440×
18,857,195×
21,016,303×
19,998,990×
17,880,098×
16,285,507×
17,510,761×
9,430,554×
5,322,400×
3,363,906×
3,098,303×
2,617,141×
1,876,242×
887,886×
925,874×
643,354×
452,684×
136,170×
221,301×
204,362×
133,238×
15,136×
32,034×
53,036×
55,723×
12,747×
22,307×
5,991×
1,116×

0s:
1s:
2s:
3s:
4s:
5s:
6s:
7s:
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47s:
48s:
49s:
50s:
51s:
52s:
53s:
54s:
55s:
56s:
57s:
58s:
0×
1×
15,351×
627,744×
5,090,518×
13,301,516×
22,127,080×
33,030,828×
41,856,968×
63,789,644×
74,270,024×
67,778,636×
60,780,132×
51,478,668×
44,980,898×
45,503,306×
42,546,398×
44,960,150×
38,054,400×
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:
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38s:
39s:
40s:
41s:
42s:
43s:
44s:
45s:
46s:
47s:
48s:
49s:
0×
1×
17,601×
1,330,251×
12,683,351×
23,215,532×
82,220,008×
134,345,156×
79,961,901×
58,052,356×
79,437,859×
110,211,195×
129,023,672×
147,678,519×
172,707,697×
201,080,120×
204,038,783×
204,653,577×
196,524,528×
181,085,065×
155,533,621×
131,658,432×
121,699,123×
120,731,132×
97,000,809×
77,733,144×
79,345,971×
74,600,300×
69,721,305×
64,101,840×
67,465,032×
51,790,177×
31,712,471×
20,550,980×
19,832,993×
17,284,585×
11,009,017×
5,422,833×
5,287,644×
2,992,939×
1,398,827×
834,185×
1,064,236×
522,243×
212,650×
37,423×
73,733×
28,684×
26,179×
3,480×

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0×
1×
21,075×
1,019,592×
9,059,168×
30,367,176×
54,228,194×
77,831,636×
94,819,602×
150,390,250×
187,624,176×
181,072,338×
203,899,808×
206,850,296×
201,149,314×
183,601,236×
150,199,920×
119,657,686×
91,229,658×
61,009,218×
48,547,164×
55,264,528×
51,449,348×
66,249,112×
87,180,892×
117,230,528×
136,736,532×
159,053,754×
142,925,286×
127,164,830×
123,969,628×
132,915,170×
139,964,900×
169,990,578×
191,645,550×
213,762,060×
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×

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0×
1×
25,755×
3,137,293×
36,244,765×
73,366,759×
290,011,625×
505,512,314×
282,035,289×
154,701,860×
173,059,477×
232,150,597×
273,740,587×
319,651,543×
386,285,807×
467,107,929×
486,200,170×
514,804,918×
520,469,725×
513,181,498×
473,456,740×
420,220,566×
391,612,727×
363,163,285×
308,148,065×
257,076,105×
267,833,957×
248,619,283×
218,224,314×
183,325,278×
192,215,341×
179,635,440×
139,230,251×
105,203,580×
96,724,213×
92,399,565×
61,638,944×
35,552,965×
30,724,453×
18,950,814×
15,833,054×
8,879,715×
7,530,913×
5,746,460×
3,469,209×
1,793,801×
1,970,283×
1,212,170×
722,660×
156,101×
309,695×
391,146×
206,874×
16,454×
64,600×
100,646×
81,644×
15,836×
44,210×
8,991×
1,500×

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0×
1×
29,175×
2,162,706×
17,270,970×
56,393,458×
108,975,308×
152,254,388×
182,001,698×
339,827,128×
431,468,512×
467,809,072×
491,230,534×
549,319,552×
556,580,182×
561,269,328×
574,184,980×
542,043,374×
470,812,672×
416,692,826×
337,531,668×
249,338,738×
190,193,596×
164,670,898×
184,186,302×
177,927,152×
183,051,970×
189,918,044×
196,461,686×
190,979,578×
184,547,484×
208,402,570×
226,271,808×
233,625,294×
214,629,256×
250,710,904×
298,563,198×
334,512,134×
331,982,288×
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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0×
1×
46,293×
7,873,217×
95,761,445×
199,998,729×
926,234,173×
1,671,386,042×
1,026,189,540×
481,379,473×
388,983,616×
471,623,136×
520,150,463×
640,598,281×
774,348,946×
982,949,905×
1,057,809,313×
1,150,092,231×
1,189,047,550×
1,199,551,415×
1,177,075,955×
1,119,215,045×
1,074,649,332×
1,074,692,588×
939,298,663×
786,550,608×
771,625,121×
764,921,216×
651,861,408×
567,748,952×
562,044,699×
544,318,035×
436,449,278×
357,866,110×
336,556,080×
299,328,308×
206,145,124×
140,810,641×
146,464,653×
117,157,551×
97,607,610×
52,513,004×
49,623,402×
35,839,738×
19,828,157×
7,712,985×
8,007,966×
4,318,280×
2,388,260×
1,142,811×
1,645,637×
761,799×
363,491×
55,686×
135,108×
48,410×
40,857×
4,323×

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0×
1×
45,591×
5,680,326×
46,411,104×
106,581,080×
195,694,206×
257,868,060×
361,843,212×
863,259,952×
1,138,104,896×
1,173,481,536×
1,130,567,794×
1,169,083,950×
1,123,533,754×
1,089,519,576×
1,137,981,898×
1,270,763,066×
1,334,876,098×
1,394,069,674×
1,391,590,730×
1,323,716,924×
1,260,180,094×
1,078,599,726×
880,175,882×
671,120,330×
524,072,220×
431,812,864×
427,609,420×
429,774,574×
368,312,680×
408,412,128×
350,986,112×
317,214,856×
229,763,858×
228,105,846×
220,619,850×
271,850,142×
313,858,228×
396,039,136×
475,665,162×
596,264,508×
678,306,310×
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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0×
1×
62,520×
15,468,431×
237,114,751×
509,938,187×
2,732,189,983×
4,958,065,073×
3,108,473,771×
1,423,227,876×
982,505,679×
978,590,104×
1,020,708,235×
1,270,336,860×
1,507,949,966×
1,911,413,118×
1,913,195,556×
2,190,323,879×
2,380,063,619×
2,582,046,328×
2,716,060,317×
2,564,748,664×
2,512,892,210×
2,589,162,584×
2,344,540,612×
2,236,208,465×
2,162,334,016×
2,083,021,065×
1,776,148,987×
1,598,052,868×
1,608,309,685×
1,394,296,723×
1,065,302,090×
982,934,971×
995,787,777×
983,660,528×
763,906,535×
515,110,746×
500,725,645×
468,064,220×
392,115,605×
308,913,155×
355,147,681×
304,263,907×
187,062,366×
113,252,243×
63,916,273×
37,040,929×
29,305,269×
15,428,922×
14,928,890×
9,475,994×
6,159,261×
2,647,786×
3,757,491×
2,215,496×
1,291,560×
209,122×
559,612×
686,575×
384,144×
23,680×
113,550×
170,080×
151,691×
19,357×
77,225×
12,503×
1,884×

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0×
1×
69,375×
12,612,130×
97,874,438×
261,608,230×
419,960,762×
593,271,816×
825,489,686×
2,095,448,466×
2,689,480,192×
2,583,861,100×
2,420,180,290×
2,504,327,914×
2,305,517,852×
1,997,977,370×
1,842,055,142×
1,932,920,990×
2,057,761,348×
2,266,730,854×
2,512,465,208×
3,074,250,176×
3,167,097,098×
3,149,856,590×
3,164,305,710×
2,842,159,532×
2,445,218,702×
2,116,029,214×
1,748,901,724×
1,373,943,266×
1,335,186,906×
1,314,271,108×
1,206,617,764×
1,114,753,238×
988,217,520×
1,047,343,040×
943,794,046×
918,048,428×
812,190,224×
729,063,432×
789,225,990×
981,116,968×
1,162,371,428×
1,478,155,884×
1,510,000,700×
1,694,926,916×
1,497,037,688×
1,389,270,792×
990,227,644×
869,286,206×
565,969,562×
464,912,654×
422,346,904×
447,785,128×
379,187,710×
446,023,892×
370,025,302×
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×