Apéry's constant is defined by
(1)
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(OEIS A002117) where is the Riemann zeta function.
B. Haible and T. Papanikolaou computed to digits using a Wilf-Zeilberger pair identity with
(2)
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, and , giving the rapidly converging
(3)
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(Amdeberhan and Zeilberger 1997). The record as of Dec. 1998 was 128 million digits, computed by S. Wedeniwski. was computed to decimal digits by E. Weisstein on Sep. 16, 2013.
The Earls sequence (starting position of copies of the digit ) for is given for , 2, ... by 10, 57, 3938, 421, 41813, 1625571, 4903435, 99713909, ... (OEIS A229074).
-constant prime occur for , 55, 109, 141, ... (OEIS A119334), corresponding to the primes 1202056903, 1202056903159594285399738161511449990764986292340498881, ... (OEIS A119333).
The starting positions of the first occurrence of , 1, 2, ... in the decimal expansion of (not including the initial 0 to the left of the decimal point) are 3, 1, 2, 10, 16, 6, 7, 23, 18, 8, ... (OEIS A229187).
Scanning the decimal expansion of until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 7, 89, 211, 2861, 43983, 292702, 8261623, ... (OEIS A036902), which end at digits 23, 457, 7839, 83054, 1256587, 13881136, 166670757, ... (OEIS A036906).
The digit sequences 0123456789 and 9876543210 do not occur in the first digits (E. Weisstein, Sep. 17, 2013).
It is not known if is normal (Bailey and Crandall 2003), but the following table giving the counts of digits in the first terms shows that the decimal digits are very uniformly distributed up to at least .
OEIS | 10 | 100 | ||||||||
0 | A000000 | 3 | 9 | 108 | 990 | 9910 | 99761 | 1000416 | 9999248 | 100001073 |
1 | A000000 | 1 | 11 | 104 | 1024 | 10037 | 100273 | 1000484 | 10000163 | 99996430 |
2 | A000000 | 2 | 9 | 109 | 1007 | 10061 | 100012 | 1001036 | 10005579 | 99985752 |
3 | A000000 | 1 | 11 | 106 | 1010 | 9961 | 99894 | 998032 | 10000695 | 100007728 |
4 | A000000 | 0 | 8 | 76 | 953 | 9957 | 99904 | 998174 | 9991603 | 99994148 |
5 | A000000 | 1 | 13 | 108 | 1006 | 9933 | 100399 | 1002043 | 10003610 | 99999279 |
6 | A000000 | 1 | 7 | 90 | 1001 | 9967 | 99525 | 999818 | 10003630 | 100014221 |
7 | A000000 | 0 | 6 | 113 | 1064 | 10253 | 100616 | 1000198 | 9995077 | 99993290 |
8 | A000000 | 0 | 12 | 90 | 981 | 9931 | 99675 | 999969 | 10001192 | 100009336 |
9 | A000000 | 1 | 14 | 96 | 964 | 9990 | 99941 | 999830 | 9999203 | 99998743 |