Glaisher-Kinkelin Constant Digits
المؤلف:
Sloane, N. J. A
المصدر:
Sequences A074962, A118420, and A225763 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
27-1-2020
1583
Glaisher-Kinkelin Constant Digits
The decimal expansion of the Glaisher-Kinkelin constant is given by
(OEIS A074962). 
was computed to 
decimal digits by E. Weisstein (Dec. 3, 2015).
The Earls sequence (starting position of 
copies of the digit 
) for 
is given for 
, 2, ... by 7, 14, 2264, 1179, 411556, ... (OEIS A225763).
The digit sequences 0123456789 and 9876543210 do not occur in the first 
digits (E. Weisstein, Dec. 3, 2015).
-constant primes occur for 7, 10, 18, 64, 71, 527, 1992, 5644, 8813, 19692, ... (OEIS A118420) decimal digits.
The starting positions of the first occurrence of 
, 1, 2, ... in the decimal expansion of 
(including the initial 1 and counting it as the first digit) are 12, 1, 2, 18, 5, 22, 14, 7, 3, 10, ... (OEIS A229193).
Scanning the decimal expansion of 
until all 
-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 5, 98, 478, 9192, ... (OEIS A000000), which end at digits 22, 495, 7233, 100426, ... (OEIS A000000).
It is not known if the Glaisher-Kinkelin constant is normal in base 10, but the following table giving the counts of digits in the first 
terms shows normal-appearing behavior up to at least 
 |
OEIS |
10 |
100 |
 |
 |
 |
| 0 |
A000000 |
0 |
11 |
96 |
999 |
9890 |
| 1 |
A000000 |
2 |
9 |
102 |
1033 |
9928 |
| 2 |
A000000 |
4 |
16 |
93 |
992 |
9977 |
| 3 |
A000000 |
0 |
8 |
100 |
1016 |
10055 |
| 4 |
A000000 |
1 |
8 |
99 |
955 |
10043 |
| 5 |
A000000 |
0 |
5 |
94 |
979 |
10034 |
| 6 |
A000000 |
0 |
12 |
96 |
988 |
10121 |
| 7 |
A000000 |
1 |
12 |
114 |
1067 |
9998 |
| 8 |
A000000 |
1 |
11 |
108 |
1031 |
9999 |
| 9 |
A000000 |
1 |
8 |
98 |
940 |
9955 |
REFERENCES:
Sloane, N. J. A. Sequences A074962, A118420, and A225763 in "The On-Line Encyclopedia of Integer Sequences."
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