Apéry,s Constant Digits
المؤلف:
Bailey, D. H. and Crandall, R. E
المصدر:
"Random Generators and Normal Numbers." Exper. Math. 11
الجزء والصفحة:
...
16-1-2020
1029
Apéry's Constant Digits
Apéry's constant is defined by
(OEIS A002117) where 
is the Riemann zeta function. 
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 |
REFERENCES:
Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.
Preprint dated Feb. 22, 2003 available at http://www.nersc.gov/~dhbailey/dhbpapers/bcnormal.pdf.
Sloane, N. J. A. Sequences A002117, A036902, A036906, A119333, A119334, A229074, and A229187 in "The On-Line Encyclopedia of Integer Sequences."
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