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The Database of Number Correlations
Pi in Other Bases
Pi is an interesting number connected with the circle. It is usually written in base 10 notation: 3.14159265358979... Many people have wondered about it's universal significance and sought deeper meanings within it's unchanging digits. Some hold the belief that Pi is the "answer to the universe,". Perhaps the deeper meaning of Pi can be better seen in some other base?
Base 2
11.00100100001111
Base 3
10.01021101222201
Base 4
3.02100333122220
Base 5
3.03232214303343
Base 6
3.05033005141512
Base 7
3.06636514320361
Base 8
3.11037552421026
Base 9
3.12418812407442
Base 10
3.14159265358979
Base 11
3.16150702865A48
Base 12
3.184809493B9186
Base 13
3.1AC1049052A2C7
14
3.1DA75CDA813752
Base 15
3.21CD1DC46C2B7A
Base 16
3.243F6A8885A300
Base e
10.10100202000211
Simple Continued Fraction Expansion of Pi: Properties and Meanings
maths.org.uk/issue11/features/cfractions
One of the interesting things about the continued-fraction expansion of (irrational) numbers is that they are, in a sense, base-independent. Instead of endlessly repeating digits of the base in which we are representing the number (digits 0 - 9 in base ten), we get "whole" numbers:
1,
2,
3, etc. - as large as you wish. So we find a relatively large "292" fairly early in the simple continued fraction form of pi. What other large numbers might there be? How common are they?
3,
7,
15,
1,
292,
1,
1,
1,
2,
1,
3,
1,
14,
2,
1,
1,
2,
2,
2,
2,
1,
84,
2,
1,
1,
15,
3,
13,
1,
4,
2,
6,
6,
99,
1,
2,
2,
6,
3,
5,
1,
1,
6,
8,
1,
7,
1,
2,
3,
7,
1,
2,
1,
1,
12,
1,
1,
1,
3,
1,
1,
8,
1,
1,
2,
1,
6,
1,
1,
5,
2,
2,
3,
1,
2,
4,
4,
16,
1,
161,
45,
1, 22,
1,
2,
2,
1,
4,
1,
2, 24,
1,
2,
1,
3,
1,
2,
1,
1,
10,
2,
5,
4,
1,
2,
2,
8,
1,
5,
2,
2,
26,
1,
4,
1,
1,
8,
2,
42,
2,
1,
7,
3,
3,
1,
1,
7,
2,
4,
9,
7,
2,
3,
1, 57,
1, 18,
1,
9,
19,
1,
2, 18,
1,
3,
7,
30,
1,
1,
1,
3,
3,
3,
1,
2,
8,
1,
1,
2,
1,
15,
1,
2,
13,
1,
2,
1,
4,
1,
12,
1,
1,
3,
3, 28,
1,
10,
3,
2, 20,
1,
1,
1,
1,
4,
1,
1,
1,
5,
3,
2,
1,
6,
1,
4,
1,
120,
2,
1,
1,
3,
1,
23,
1,
15,
1,
3,
7,
1,
16,
1,
2,
1,
21,
2,
1,
1,
2,
9,
1,
6,
4,
127,
14,
5,
1,
3,
13,
7,
9,
1,
1,
1,
1,
1,
5,
4,
1,
1,
3,
1,
1, 29,
3,
1,
1,
2,
2,
1,
3,
1,
1,
1,
3,
1,
1,
10,
3,
1,
3,
1,
2,
1,
12,
1,
4,
1,
1,
1,
1,
7,
1,
1,
2,
1,
11,
3,
1,
7,
1,
4,
1, 48,
16,
1,
4,
5,
2,
1,
1,
4,
3,
1,
2,
3,
1,
2,
2,
1,
2,
5, 20,
1,
1,
5,
4,
1,
436,
8,
1,
2,
2,
1,
1,
1,
1,
1,
5,
1,
2,
1,
3,
6,
11,
4,
3,
1,
1,
1,
2,
5,
4,
6,
9,
1,
5,
1,
5,
15,
1,
11, 24,
4,
4,
5,
2,
1,
4,
1,
6,
1,
1,
1,
4,
3,
2,
2,
1,
1,
2,
1, 58,
5,
1,
2,
1,
2,
1,
1,
2,
2,
7,
1,
15,
1,
4,
8,
1,
1,
4,
2,
1,
1,
1,
3,
1,
1,
1,
2,
1,
1,
1,
1,
1,
9,
1,
4,
3,
15,
1,
2,
1,
13,
1,
1,
1,
3, 24,
1,
2,
4,
10,
5,
12,
3,
3, 21,
1,
2,
1, 34,
1,
1,
1,
4,
15,
1,
4, 44,
1,
4, 20776,
1,
1,
1,
1,
1,
1,
1, 23,
1,
7,
2,
1,
94,
55,
1,
1,
2,
1,
1,
3,
1,
1,
32,
5,
1,
14,
1,
1,
1,
1,
1,
3,
50,
2,
16,
5,
1,
2,
1,
4,
6,
3,
1,
3,
3,
1,
2,
2,
2,
5,
2,
2,
2, 28,
1,
1,
13,
1,
5, 43,
1,
4,
3,
5,
3,
1,
4,
1,
1,
2,
2,
1,
1, 19,
2,
7,
1, 72,
3,
1,
2,
3,
7,
11,
1,
2,
1,
1,
2,
2,
1,
1,
2,
1,
1,
1,
1,
1,
33,
7,
19,
1,
19,
3,
1,
4,
1,
1,
1,
1,
2,
3,
1,
3,
2,
2,
2,
2,
4,
1,
1,
1,
4,
2,
3,
1,
1,
1,
1,
11,
1,
1,
2,
1,
2,
1,
2,
2,
1,
7,
2,
27,
1,
1,
6,
2,
1,
9,
6,
26,
1,
1,
3,
2,
1,
1,
1,
1,
1,
15,
1, 36,
4,
2,
2,
1, 22,
2,
1,
106,
2,
2,
1,
3,
1,
12,
10,
7,
1,
2,
1,
1,
1,
1,
8,
2,
4,
5,
3,
2,
1,
4, 23,
1, 18,
2,
10,
3,
1,
6,
6,
13,
8,
6,
2,
2,
2,
2,
1,
1,
1,
3,
1,
7,
17,
1,
1,
1,
2,
5,
5,
1,
1,
2,
11,
1,
6,
1,
6,
1, 29,
4, 29,
3,
5,
3,
1,
141,
1,
2,
7,
7,
2,
2,
7,
1,
1,
7,
1,
7,
1,
2,
4,
1,
1,
1, 30,
1,
12,
4, 18,
10,
2,
8,
1,
2,
2,
2,
4,
13,
1,
5,
4,
1,
6,
1,
1,
11,
2,
4,
2,
1,
1,
3,
3,
12,
1,
1, 39,
5,
1,
1,
16, 125,
1,
4,
1,
2,
1, 19,
1,
4,
1,
1,
2,
1,
4,
1,
10,
1,
4,
2,
1,
1,
1,
5,
10,
4,
14,
1,
13, 41,
1,
4,
1,
8,
1,
1,
2,
1,
3,
1,
6,
1,
3,
2,
2,
2,
1,
4,
1,
14,
1,
2,
8,
1,
8,
3,
3,
3,
1, 37,
4,
2,
4,
1,
3,
4, 25,
4, 27,
2,
7,
1,
1,
2,
6,
1,
1,
1,
12,
1,
2,
2,
2,
13,
12,
1,
3,
1,
6,
1,
1, 33,
1,
5,
3,
1,
5,
15,
8,
8, 47,
1,
3,
2,
12,
2,
12,
1,
12,
1,
2,
5,
3,
1,
1,
1,
1,
2,
3,
5,
4,
2,
1,
1,
5,
1,
9,
14,
1,
1,
3,
2,
1,
9,
3, 22,
13,
1,
1,
3, 20,
1,
1, 61,
1, 376,
2, 107,
1,
10,
3,
2,
2, 31,
1,
2,
10,
2,
2, 62,
2,
2,
7,
4,
5,
6,
1,
1,
1,
1,
2,
8,
2, 73,
3,
5, 42,
1,
3,
2,
1,
1, 59,
6,
1,
1,
1,
5,
1,
6,
1,
2,
6,
1,
1,
1,
1,
3,
2,
1,
3,
1,
8,
1,
4,
2,
5,
4,
7,
1,
4,
2,
2,
6,
1,
1,
2,
2,
1,
1,
1,
1,
1,
2,
1,
2,
2,
5,
1,
2,
1,
1,
10,
1,
6,
1, 129,
1,
4, 65,
2,
4,
4,
3,
2,
3,
1,
1,
5,
1,
1,
1,
1,
1,
2,
2,
1,
2,
1,
1,
2,
2,
1,
2,
3,
1,
2,
1,
2,
4,
2,
1,
2, 27,
6,
2,
1, 193,
1,
3,
9,
1,
3, 35,
2,
1,
8,
1,
1,
1,
1,
9,
3, 56,
1,
6,
6,
2,
8,
1,
8,
1,
2,
3,
6,
3,
1,
3,
1,
1,
1,
2,
13,
1,
1,
1,
1,
13,
2,
1,
3,
1,
3,
15,
2,
1,
1,
2,
4,
1,
4,
5,
2,
2,
1,
2,
1,
6,
1,
4,
12,
1,
1,
1,
1,
13,
1,
3,
4,
1,
1,
1,
2,
9,
1,
7,
1,
1,
1,
1,
4,
1,
3,
4,
1,
1,
4,
3,
1, 39,
2,
1,
1,
1,
1,
1,
4,
7,
2,
2,
2,
1,
1,
1,
1,
2, 114,
12,
4,
1,
3,
2,
1, 19,
1,
1,
2,
1,
1,
3,
4,
1, 60,
3, 72,
2,
1,
1,
1, 50,
1,
1,
1,
1,
3,
1,
1,
2,
2,
1,
4,
1,
7,
3,
1,
2,
1,
5,
1,
1,
1,
2,
6,
2, 21,
2,
6,
1,
6,
1,
1,
2,
1,
7,
1,
8,
1,
1,
5,
4,
1,
1,
1,
1,
1,
1,
1,
1,
4,
1,
11,
2,
4,
10,
2,
1,
1,
13,
1,
1,
7,
15,
1,
1,
1,
2,
3,
15,
8,
8,
2,
1,
13,
3,
5,
1,
2,
1,
6,
1,
10,
123,
3,
1,
4,
59,
4,
156,
88,
1,
5,
4,
1,
3,
1,
4,
2,
9,
1,
7,
4,
2,
1,
2,
3,
2,
1,
2,
11,
1,
13,
7,
7,
1, 63, 37,
12, 86,
1,
1,
1,
1,
2,
2,
4,
2, 18,
1,
1,
1, 41,
2,
1,
1,
12,
1,
2,
1,
1,
2,
10,
1,
1,
1,
5,
1,
1,
3,
1,
7,
5,
1,
9,
1,
2,
2,
7,
1,
1,
5,
2,
1,
3,
3,
5,
2,
1,
11,
3,
1,
3,
2,
1,
1,
2,
1,
14,
5,
2,
2,
1,
1,
1,
1,
3,
1,
3,
3,
2,
2,
1,
3,
2,
1,
2,
1,
4,
1,
14,
1,
1,
58,
7,
1,
2,
1,
1,
5,
1,
2,
1,
5, 18,
1,
4,
3,
1,
1,
1,
4,
1,
1,
2,
5,
1, 148,
1,
9,
2,
1,
2,
1,
5,
4, 93,
1,
1,
2,
4,
1,
2, 73,
1,
1,
3,
1,
1,
1,
1,
2,
1, 34,
1,
5,
6,
1,
2,
1,
3,
4,
1,
16, 28,
17,
2,
5,
5, 26,
1,
1,
4,
12,
1,
3,
2,
1,
5,
1,
2,
9,
3,
2, 41,
1,
16,
2,
2, 20,
1,
17,
1,
6,
16,
3,
3,
2,
2,
2, 18,
15,
1,
1, 51,
4,
9,
5,
2,
2,
1,
2,
1, 45,
3,
1,
1,
3,
1,
2,
1,
3,
1,
1,
3,
5,
1,
2,
3,
8,
2, 47,
2,
3,
1,
1,
1,
15,
9,
1,
8,
2,
1,
4,
2,
4,
14,
1,
12,
2,
1, 161,
1, 26,
2,
1,
2,
1,
1,
1,
1,
2,
2,
1, 18, 528,
12,
4,
1,
5,
16,
3,
1,
1,
1,
1,
1,
5,
1,
2,
1, 63,
1, 97,
1,
4,
4,
10,
5,
9,
5,
2,
3,
2,
5,
7,
1, 32,
13,
1,
5,
4,
1,
7,
1,
3,
12,
1,
3,
9,
1,
7,
1, 102, 53,
1,
1,
1,
3,
4,
2,
15,
2,
8,
2,
2,
3,
1,
2,
4,
1,
1,
3,
2,
3,
1,
1,
2,
3,
1,
1,
6,
1,
1,
14,
1, 80,
11,
1,
1,
1,
1, 22,
1,
2,
3,
1,
3, 26,
2, 24,
2,
2,
4,
3,
1,
1,
1,
1,
3,
1, 63,
1,
1,
1, 25,
1,
1,
1,
8,
1,
3,
3,
1,
10,
5,
6,
2,
1,
1,
3,
1,
1,
1,
1,
2,
2,
1,
2,
8,
12,
1, 53,
1,
2,
1,
1,
5,
1,
1,
3,
1, 39,
1,
12,
1,
3,
14, 18,
9,
3,
2,
2,
2,
1,
1,
3,
1,
4,
4,
7,
1,
17,
1,
14,
1,
1,
1,
1,
3,
1,
1,
10,
1,
2,
2,
3,
1,
2,
1,
2,
2,
2,
12,
1,
3, 44,
2,
10,
1,
14,
1,
2,
1, 43,
4,
1,
7,
3,
4,
1,
1,
2,
2,
1, 34,
1,
2,
5,
8,
3,
2,
1,
2,
13,
4,
3,
2,
1,
1,
1,
1, 25,
1,
5,
1, 94,
2,
4,
3,
4,
5,
1,
1,
1,
1,
1,
1,
2,
1,
1,
1,
1,
1,
1,
1,
1,
1,
10, 41,
1,
5,
1,
4,
4,
1, 155,
1,
8,
1,
1,
1,
1,
4,
1,
1,
2,
9,
2,
1,
2,
1,
1,
1,
6, 23,
1,
2,
3,
5,
2,
1,
1,
1,
1,
7, 67,
5,
7,
1, 23,
3,
3,
1,
6,
1,
11,
1, 57,
1,
4,
1,
5,
1,
1,
8,
1,
1,
2,
5,
2,
10,
1,
1,
2,
1,
1,
3,
1,
2,
1,
3,
1,
11,
2,
10,
1,
4, 18,
1,
2,
3,
1,
1,
6,
3,
6,
4, 31,
3,
4,
1, 18,
3,
9,
7,
5,
1,
2,
2,
1,
7,
1, 23,
2, 217,
1,
2,
1,
4,
1, 54,
2, 196,
10,
3,
1, 32,
1, 40, 55,
1,
5,
1,
3,
3,
1,
2,
2,
1,
3,
6,
3,
16,
1, 31,
1,
5,
6,
1,
4, 42,
4,
1,
10,
1,
3,
1,
3,
3,
1,
2,
1,
1,
1,
4,
1,
13,
1, 88,
1,
1,
1,
14,
3, 27,
3,
1,
1,
16,
4,
1,
2,
4,
1,
4,
1,
1,
17,
2,
4,
1,
1,
9,
2,
1,
1,
3,
1,
1, 30,
1,
1,
3,
2,
2,
1,
1,
4,
10,
1,
7,
1,
6,
1, 35,
1,
1,
2,
3,
6,
1,
1,
2,
4,
4, 24,
1,
1,
1,
1,
1,
1,
3,
1,
1,
2,
1,
2,
1,
6,
6,
2,
1,
1,
10,
6,
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4,
3,
1,
5,
2,
5,
1,
2,
1, 75, 22,
1,
1, 21,
1,
5,
9,
1,
1,
2,
3,
6,
14,
1,
2,
3,
3,
2,
2, 756, 26,
1,
3,
2,
1,
4,
1,
1, 265,
4,
2,
2, 37,
6,
1,
1,
13,
5,
2,
1,
6,
2,
1,
6,
6, 46,
15,
1,
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2,
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2, 21,
4,
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8,
4,
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3,
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4,
7,
2,
1,
7,
3,
7,
4,
2,
2,
2,
6,
2,
1,
1,
1,
5,
5,
1, 94,
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2,
2,
3,
11,
1,
1,
7,
1,
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2,
1,
4,
1,
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6,
1,
7,
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2,
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1,
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7,
2,
3,
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2,
3,
1,
5,
6,
11,
2,
40,
7,
1,
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29,
3,
2,
534,
6,
1,
3,
2,
2,
1,
2,
2,
1,
9,
1,
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3,
6,
1,
1,
6,
4,
55,
45,
3,
1,
8,
2,
3,
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1,
5,
1,
88,
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1,
1,
1,
1,
2,
2,
1,
13,
1,
3,
5,
2,
2,
2, 21,
1,
1,
11,
9,
2,
2, 26,
1,
1,
1,
1,
76,
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2,
1,
1,
3,
1,
7,
109,
3,
1,
3,
5,
1,
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3,
1,
1,
7,
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6,
8,
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8,
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1,
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3, 24,
4,
2,
4,
2,
2,
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7,
1,
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8,
1,
6, 182,
1,
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4,
5,
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4,
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2,
5,
1,
7,
2, 29,
3,
13,
8,
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6,
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1,
1,
4,
4, 59,
1, 27,
4,
4,
2,
7,
2,
3,
1,
1,
4,
3,
2,
8,
4,
3,
4,
3,
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1,
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11,
1,
1,
8,
1,
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2,
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7,
1,
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1,
8,
1,
7,
1,
4,
2,
1,
2,
1,
3,
1,
8,
1,
6,
2, 632,
1,
1,
1,
5,
10,
1,
6,
5,
1,
5,
1,
1,
1,
1,
1,
1,
1, 97,
2,
2,
1,
3,
1,
3,
1,
1, 72, 58,
1,
2,
1,
1,
1, 21, 19,
1,
8,
3,
1,
1,
1,
2, 41,
1, 22,
4,
1,
1,
1,
1,
6,
8,
1,
1, 18,
1, 33,
2, 537,
3,
1,
2,
4,
10,
2, 53,
2,
4,
1,
6,
1,
3,
13,
1,
1, 50,
2,
1,
1,
2,
3,
1,
3,
6,
1, 29,
1,
2,
1,
1,
5,
1,
6,
2,
2,
1,
2,
3,
1,
11,
2,
1,
4,
12,
1,
1,
1,
1,
4,
1,
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2,
3,
2,
3,
2,
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4,
1,
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3,
2,
2,
1,
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2,
2,
2,
1,
7,
6,
1,
4,
1,
1,
2,
1,
2,
1,
4,
1,
4,
4,
1,
1,
1,
3,
2,
8,
5, 134,
11,
2,
1,
12,
3,
2,
4,
1,
13,
2,
2,
2, 25, 25,
4,
1,
2,
14,
1,
1,
1,
9,
1,
1,
3,
1,
4,
2,
1,
1,
1,
1,
1,
2,
6,
1, 69, 100,
1,
5,
4,
2,
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1,
15,
12,
16,
3,
1,
1,
7,
1,
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2,
4,
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2,
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1,
6,
2,
9,
1,
2,
3,
6,
1,
1,
4,
1,
2,
1, 62,
1,
6,
4,
1, 24,
8,
3,
2,
10, 26,
1,
8,
1,
1,
12,
1,
6,
3,
1,
2,
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2,
1,
2,
1,
4,
11,
5,
1,
1,
2,
1,
8, 24,
2,
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5,
14,
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3,
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8,
1,
1,
2,
1,
1,
4,
2,
1,
3,
5,
2,
1,
5,
3,
5,
2, 138,
1,
9,
2,
1,
2,
1,
9,
1,
2,
1,
1,
1,
4,
1,
1,
2,
4, 27,
2,
1,
7,
2,
1,
9,
1,
1,
1,
1,
2,
2,
1,
2,
3,
1,
3,
2,
1,
4,
5,
1,
3,
1,
5,
2,
1,
1,
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6,
1,
4,
2,
1,
1,
5,
1,
1,
1,
5,
1,
3,
5,
1,
2,
3,
3,
1,
1,
3,
1,
1,
7,
2,
1,
1,
5,
2,
3,
5,
1,
2,
1,
7,
2,
1,
2,
1,
1,
5,
1,
5,
1,
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2,
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1,
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3,
2,
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3,
1,
3,
1,
3,
7,
2,
11,
10,
1,
1,
1,
1,
6,
4,
4,
1,
17,
10,
1,
1,
2,
7,
2,
1,
1,
1,
3,
2,
1,
2,
4,
1,
4, 55,
1,
5,
1,
1,
3,
1,
5,
2,
8,
1,
2,
2,
1,
11,
1,
3,
2,
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4,
8,
3,
2,
1,
1,
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3,
1,
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7,
1,
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2,
1,
7,
1,
6,
1,
3,
1,
3,
1,
2,
1,
1,
1,
4,
1,
1, 42,
2,
1,
13,
1,
1,
1, 626,
5,
2,
2,
7,
2,
4,
1, 102,
2,
12,
1,
1,
2,
1,
1,
2,
5,
1,
3,
1,
2,
3,
1,
2,
1,
6,
1,
1,
1,
1,
15,
1,
1,
2,
4,
1,
1,
4,
3,
10,
2,
9,
4,
1, 211,
3,
1, 34,
1,
4,
2,
1,
1,
1,
4,
1,
12,
1,
3,
1,
2,
4,
2...
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