Base 12: Iteration Frequencies: Full Cycle

Blog: The First Pixel: Kaprekar's Constant 6174

Radix Character Encoding
0123456789AB

Color Key
OrangeFixed Pointnumber that becomes itself after one iteration
GreenFinal Numbersall numbers eventually reached, from all cycles
PurpleCycle Startiterations include entry into cycle, assuming it repeats
RedFull Cycleiterations include entire cycle, proving it repeats
BlueStart Numberssubset (optimization) of all numbers; minimal for full coverage
GrayNo Resultsany calculation that has no results

Digits 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33
Average Steps
Full Cycle
1.92 step
avg
6.56 step
avg
4.69 step
avg
8.31 step
avg
7.68 step
avg
17.89 step
avg
19.49 step
avg
30.10 step
avg
18.37 step
avg
17.14 step
avg
23.83 step
avg
13.71 step
avg
21.23 step
avg
12.98 step
avg
19.85 step
avg
11.89 step
avg
22.06 step
avg
11.01 step
avg
26.84 step
avg
11.16 step
avg
29.70 step
avg
10.97 step
avg
36.32 step
avg
12.63 step
avg
41.44 step
avg
11.95 step
avg
44.75 step
avg
12.19 step
avg
46.33 step
avg
14.27 step
avg
43.74 step
avg
16.38 step
avg
39.99 step
avg
Step Frequency
Full Cycle
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2s:


11×
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11×




33×
30×
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7s:
8s:


53×
40×
78×
70×
58×
42×
22×
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13s:


11×

82×
82×
62×
209×
241×
302×
174×
83×
82×
36×
0s:
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111×
272×
328×
422×
502×
382×
426×
420×
594×
658×
230×
22×
0s:
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53×








500×
920×
1,314×
823×
1,139×
728×
857×
1,134×
770×
501×
399×
530×
651×
392×
599×
525×
396×
124×
20×
0s:
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203×
388×
422×
562×




1,294×
1,884×
2,700×
1,906×
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:
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48s:


291×
924×
1,392×
884×
1,023×
296×
441×
505×
370×
118×
74×
62×
174×
82×









1,916×
6,092×
7,339×
6,284×
4,371×
3,433×
4,332×
3,679×
4,122×
3,677×
2,990×
3,397×
4,535×
3,115×
2,866×
2,421×
1,199×
435×
576×
846×
388×
444×
454×
34×
0s:
1s:
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31s:


53×



2,162×
4,274×
5,636×
3,236×
2,926×
3,772×
1,768×
1,216×
3,588×
9,912×
14,004×
16,546×
18,326×
18,050×
14,150×
8,146×
7,030×
4,334×
3,762×
5,552×
4,010×
3,884×
4,944×
2,880×
2,804×
994×
0s:
1s:
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40s:


111×
160×
476×
3,704×
6,523×
16,295×
26,755×
22,110×
22,395×
26,419×
18,928×
20,315×
14,162×
12,955×
10,879×
8,982×
10,817×
11,303×
9,747×
8,097×
6,420×
7,197×
8,629×
8,385×
3,757×
5,495×
6,116×
5,243×
5,289×
9,062×
10,655×
7,910×
5,264×
3,909×
2,390×
2,609×
1,960×
972×
320×
0s:
1s:
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657×
3,624×
8,122×
9,320×
12,912×
10,452×
14,450×
14,478×
10,168×
10,030×
10,530×
13,544×
16,180×
19,054×
18,792×
14,902×
6,720×
4,098×
4,994×
10,082×
23,960×
40,328×
36,636×
45,094×
36,928×
41,750×
41,248×
35,328×
41,942×
27,182×
19,732×
21,690×
25,928×
11,256×
4,272×
2,552×
1,536×
7,508×
6,712×
2,236×
1,408×
2,718×
5,996×
7,078×
1,304×
0s:
1s:
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36s:
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38s:


296×
2,700×
9,152×
4,048×
14,213×
65,361×
142,092×
145,761×
123,660×
105,205×
75,380×
79,639×
75,610×
74,434×
61,879×
45,680×
46,839×
38,862×
38,224×
41,715×
39,045×
37,208×
23,848×
17,725×
15,494×
9,850×
3,984×
2,944×
1,256×
782×
1,534×
1,892×
534×
887×
1,448×
2,120×
776×
0s:
1s:
2s:
3s:
4s:
5s:
6s:
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42s:
43s:
44s:
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46s:


171×
260×



2,388×
30,774×
86,504×
84,060×
108,848×
105,198×
151,152×
183,390×
207,152×
193,494×
114,654×
138,796×
102,230×
107,122×
83,300×
65,324×
29,170×
8,526×
4,596×
11,294×
1,434×





12,536×
86,526×
124,596×
124,012×
103,422×
84,264×
56,544×
40,684×
20,880×
11,244×
7,652×
2,960×
614×
372×
0s:
1s:
2s:
3s:
4s:
5s:
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32s:
33s:
34s:
35s:
36s:
37s:
38s:
39s:


1,038×
7,510×
1,818×

8,871×
232,838×
611,343×
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:
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31s:
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36s:
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38s:
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40s:


153×
320×
1,776×
27,798×
76,956×
87,888×
84,056×
106,792×
105,060×
206,720×
224,056×
286,368×
360,192×
454,348×
479,622×
558,892×
497,326×
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:
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29s:
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34s:
35s:
36s:
37s:
38s:


414×
3,120×


21,700×
864,930×
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:
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42s:
43s:
44s:
45s:
46s:


1,035×
25,456×
42,472×
22,942×
3,966×
1,644×

13,552×
340,984×
1,046,384×
1,103,878×
1,068,692×
1,273,418×
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:
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38s:


1,752×
58,200×
157,549×
206,990×
319,454×
2,850,388×
6,212,462×
5,401,648×
4,063,025×
3,303,936×
2,687,807×
2,413,913×
1,626,177×
1,223,290×
861,557×
786,074×
614,451×
532,465×
334,946×
277,367×
206,798×
153,337×
94,704×
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:
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51s:
52s:
53s:


1,035×
25,200×
4,750×
101,292×
368,720×
367,202×
362,522×
220,112×
166,772×
71,466×
12,700×
1,678×
10,490×
379,128×
1,205,036×
1,315,398×
1,708,234×
2,369,168×
2,652,720×
2,703,574×
2,541,132×
2,391,888×
2,517,708×
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:
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31s:
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34s:
35s:
36s:
37s:
38s:


1,666×
42,270×
135,192×
208,302×
361,451×
7,852,990×
16,167,870×
13,925,042×
9,319,654×
6,906,551×
5,568,459×
5,142,965×
4,275,559×
3,100,045×
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:
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63s:


2,133×
39,948×
4,956×



8,832×
432,102×
1,425,788×
1,446,914×
960,616×
548,814×
228,194×
1,045,200×
3,629,074×
3,182,726×
3,564,612×
2,998,650×
2,761,306×
3,293,192×
3,140,536×
3,605,320×
4,942,060×
5,257,392×
5,374,776×
4,785,596×
4,962,132×
5,500,956×
5,511,250×
4,667,752×
4,502,218×
4,698,800×
6,423,578×
5,681,258×
4,709,992×
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:
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1,143×
109,160×
685,544×
1,230,062×
1,512,154×
19,888,599×
38,370,743×
31,435,964×
20,522,717×
15,067,978×
12,093,406×
10,684,392×
9,527,817×
7,249,508×
5,408,339×
5,002,848×
4,301,331×
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×
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3,243×
226,776×
1,042,036×
780,462×
1,173,382×
1,131,424×
568,388×
657,462×
184,760×
24,814×
2,654×











49,648×
3,501,502×
11,621,864×
11,619,938×
13,575,376×
14,654,844×
13,285,076×
13,606,344×
11,858,638×
10,969,276×
12,972,538×
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×
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1,851×
155,246×
431,500×
310,318×
432,890×
43,913,260×
84,254,316×
66,998,696×
39,961,185×
27,058,732×
21,188,413×
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×
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4,047×
232,282×
376,618×
421,752×
715,490×
1,414,600×
7,513,886×
5,786,234×
6,763,148×
7,223,258×
5,579,230×
6,716,830×
5,120,620×
4,913,586×
4,310,700×
3,312,466×
2,436,016×
2,276,658×
2,243,670×
2,110,630×
1,633,894×
1,168,916×
758,562×
164,576×
84,158×
43,590×
11,992×
29,336×
2,157,344×
4,814,960×
4,728,948×
6,888,306×
10,944,676×
16,177,996×
27,425,166×
24,098,352×
24,887,916×
25,685,002×
23,034,484×
21,524,046×
20,640,282×
20,866,688×
25,199,396×
22,168,996×
21,252,228×
20,824,838×
20,125,482×
24,064,552×
25,207,352×
21,173,798×
16,170,160×
12,107,514×
12,151,466×
10,396,182×
8,790,504×
9,540,298×
7,786,994×
9,118,514×
8,665,112×
5,616,386×
3,713,818×
2,721,034×
1,797,172×
1,271,820×
950,784×
850,206×
646,974×
276,114×
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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817×
68,764×
147,574×
695,456×
3,516,878×
101,577,762×
185,068,018×
146,202,417×
82,972,913×
50,308,493×
35,506,788×
28,154,042×
22,774,647×
19,010,687×
13,949,944×
15,233,429×
14,502,324×
14,030,872×
15,239,408×
16,881,547×
18,760,087×
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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4,661×
281,442×
726,466×
936,500×
1,365,416×
5,745,928×
31,038,116×
27,599,760×
26,730,726×
28,111,940×
26,331,634×
29,353,982×
28,978,966×
25,181,694×
19,528,710×
13,264,284×
8,460,044×
6,672,830×
5,444,672×
5,271,140×
3,392,880×
2,088,992×
1,373,030×
376,086×
154,838×
55,054×
8,460×





27,984×
3,174,800×
6,714,330×
6,324,836×
8,945,654×
14,292,018×
15,804,626×
15,268,188×
13,800,098×
15,034,952×
19,228,384×
28,402,212×
40,958,612×
36,671,250×
38,351,716×
40,554,500×
34,291,368×
33,061,190×
34,213,284×
30,219,528×
31,180,042×
28,861,640×
25,206,608×
23,958,896×
24,769,582×
34,565,574×
40,316,340×
32,200,272×
22,110,308×
17,094,966×
16,923,368×
13,666,522×
13,034,106×
16,040,286×
14,956,344×
17,471,522×
18,252,408×
15,304,588×
11,813,848×
7,309,516×
5,116,716×
3,593,588×
3,593,538×
4,267,218×
4,372,158×
6,223,472×
7,498,260×
4,097,806×
3,064,886×
1,293,914×
498,340×
353,538×
216,750×
145,286×
49,882×
18,772×
17,188×
18,236×
17,544×
10,678×
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4,273×
614,100×
1,823,737×
1,575,708×
1,464,741×
192,966,576×
336,458,470×
243,897,598×
127,772,780×
92,402,721×
82,326,530×
81,201,069×
75,967,399×
65,397,948×
46,657,140×
45,084,750×
35,118,073×
27,972,221×
27,050,985×
32,083,620×
37,909,143×
35,746,106×
26,363,344×
17,055,831×
12,798,582×
10,725,645×
4,956,867×
2,212,275×
1,689,520×
1,420,083×
1,589,010×
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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5,259×
410,340×
1,159,028×
2,424,102×
4,329,334×
22,907,516×
105,050,866×
95,938,566×
78,232,592×
85,101,688×
74,501,426×
87,406,100×
79,767,040×
72,744,860×
53,101,516×
31,929,986×
15,940,602×
10,455,646×
7,443,024×
8,296,658×
4,870,330×
2,499,310×
1,383,516×
411,132×
93,504×
51,386×
948×










32,368×
4,506,106×
9,265,378×
8,112,508×
12,417,360×
18,357,696×
20,379,550×
21,229,252×
18,700,630×
18,527,910×
23,121,996×
24,996,922×
20,224,768×
19,820,878×
23,431,976×
31,945,232×
40,728,788×
56,262,454×
54,296,544×
52,857,520×
50,975,012×
46,038,650×
40,997,718×
38,348,226×
36,302,794×
37,572,406×
35,389,186×
29,870,210×
32,487,118×
36,504,052×
47,932,724×
52,762,966×
44,364,526×
34,446,422×
23,937,650×
24,035,886×
18,541,336×
18,741,568×
19,531,214×
20,361,972×
30,374,184×
38,693,012×
31,210,606×
20,460,148×
14,773,342×
12,320,220×
7,883,154×
8,491,308×
11,714,196×
11,867,782×
15,079,088×
20,591,262×
16,114,720×
11,154,832×
6,467,438×
4,271,464×
2,425,390×
3,047,210×
3,602,742×
4,005,508×
6,761,410×
7,122,750×
4,851,174×
2,301,184×
1,004,568×
269,112×
115,282×
13,674×
932×
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2,335×
306,510×
365,270×
1,147,264×
7,946,298×
386,231,564×
636,495,045×
443,328,327×
213,729,006×
121,316,525×
85,476,915×
69,659,093×
55,652,492×
55,394,140×
60,926,298×
69,491,642×
78,705,947×
91,199,821×
90,517,472×
77,657,060×
81,815,358×
70,166,674×
50,976,580×
39,308,417×
32,656,998×
36,057,196×
37,946,093×
34,331,425×
32,822,128×
28,856,392×
24,169,516×
19,142,261×
15,178,289×
14,674,194×
17,934,987×
27,041,614×
18,026,964×
12,426,225×
8,785,271×
4,679,707×
3,008,986×
1,695,351×
568,472×
306,594×
387,682×
473,734×
142,144×
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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4,395×
337,814×
1,289,860×
3,464,272×
6,035,380×
57,464,246×
281,755,592×
256,053,352×
197,320,276×
206,808,660×
192,427,544×
228,934,912×
219,218,548×
204,553,830×
144,992,562×
78,366,372×
39,590,064×
28,841,768×
18,365,622×
16,673,946×
9,853,496×
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36,752×
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1,020,140×
391,698×
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294,878×
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574×
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4,324×
1,313,360×
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740,353,662×
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68,586,615×
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44,078,117×
44,445,687×
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20,548,077×
14,493,706×
8,666,946×
6,710,404×
12,744,678×
12,649,707×
13,179,574×
7,703,462×
5,053,512×
4,908,372×
4,491,280×
8,279,004×
9,647,636×
8,169,990×
6,750,444×
2,463,267×
900,720×
401,707×
93,666×
12,874×
15,992×
11,283×
1,936×
1,070×
1,182×
1,222×
935×
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7,355×
1,604,578×
11,031,026×
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16,300,276×
151,200,278×
650,154,492×
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440,122,284×
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491,823,260×
434,944,222×
403,945,968×
294,763,080×
151,030,150×
78,446,686×
69,252,540×
67,774,070×
37,973,086×
18,186,482×
7,410,684×
1,469,346×
314,782×
128,142×
76,008×
390×




















41,136×
8,163,906×
14,519,940×
12,156,088×
19,869,824×
28,188,266×
29,430,360×
30,922,534×
30,651,372×
29,744,806×
38,014,994×
38,603,118×
29,233,210×
32,951,446×
37,823,886×
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38,931,022×
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40,558,352×
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37,490,714×
46,433,236×
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82,274,732×
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31,598,362×
31,487,016×
40,653,826×
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61,734,162×
46,839,676×
39,729,424×
33,095,338×
23,815,412×
16,966,256×
19,747,032×
21,997,692×
27,390,488×
38,796,112×
42,307,192×
35,991,060×
29,393,898×
18,049,890×
14,669,164×
8,265,858×
13,635,780×
17,294,940×
16,011,282×
26,491,730×
26,226,422×
23,690,870×
21,151,404×
12,534,054×
9,535,210×
4,449,168×
7,644,752×
9,160,308×
10,659,396×
16,300,034×
15,884,022×
10,893,082×
11,976,936×
7,597,934×
3,867,890×
1,988,302×
2,775,244×
3,507,562×
4,549,018×
7,058,832×
5,976,460×
4,225,940×
1,852,784×
778,936×
145,262×
131,954×
17,172×
1,172×