Natural Logarithm of 10 Digits
المؤلف:
Sloane, N. J. A.
المصدر:
Sequences A002392/M0394, A228240,A228243, and A229197 in "The On-Line Encyclopedia of Integer Sequences."
الجزء والصفحة:
...
29-1-2020
1576
Natural Logarithm of 10 Digits
The numerical value of 
is given by
(OEIS A002392). It was computed to 
decimal digits by S. Kondo on May 20, 2011 (Yee).
The Earls sequence (starting position of 
copies of the digit 
) for 
is given for 
, 2, ... by 20, 111, 56, 9041, 4767, 674596, 24611354, 64653957, 131278082, ... (OEIS A228243).
-constant primes occur at 1, 2, 40, 242, 842, 1541, 75067, ... decimal digits (OEIS A228240).
The starting positions of the first occurrence of 
, 1, ... in the decimal expansion of 
(including the initial 2 and counting it as the first digit) are 3, 21, 1, 2, 13, 5, 17, 22, ... (OEIS A229197).
Scanning the decimal expansion of 
until all 
-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 22, 701, 7486, 88092, 1189434, 13426407, ... (OEIS A229124), which end at digits 7, 38, 351, 8493, 33058, 362945, ... (OEIS A229126).
The digit strings 0123456789 first occurs starting at position 3349545080, but 9876543210 does not occur in the first 
decimal digits of 
(E. Weisstein, Aug. 27, 2013).
It is not known if 
is normal, 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 |
2 |
14 |
103 |
964 |
9979 |
99887 |
1000356 |
9996731 |
99987842 |
1000052045 |
| 1 |
A000000 |
0 |
5 |
82 |
1039 |
10088 |
100429 |
998340 |
9999245 |
100008060 |
1000031412 |
| 2 |
A000000 |
2 |
11 |
106 |
1008 |
9943 |
99927 |
1000879 |
10005133 |
100013939 |
1000008327 |
| 3 |
A000000 |
1 |
8 |
81 |
1045 |
10108 |
100166 |
999329 |
10001639 |
100010112 |
999996738 |
| 4 |
A000000 |
0 |
9 |
106 |
973 |
10006 |
100210 |
998402 |
9999948 |
99988853 |
999996641 |
| 5 |
A000000 |
2 |
9 |
103 |
967 |
9961 |
99375 |
1001381 |
10000670 |
100011829 |
999956649 |
| 6 |
A000000 |
0 |
9 |
107 |
967 |
9828 |
99769 |
999489 |
10001182 |
99972292 |
999972639 |
| 7 |
A000000 |
0 |
11 |
110 |
1007 |
10061 |
99782 |
1000327 |
9999623 |
99989099 |
999983529 |
| 8 |
A000000 |
1 |
10 |
106 |
1000 |
9987 |
100633 |
1001812 |
9998524 |
100004534 |
999974468 |
| 9 |
A000000 |
2 |
14 |
96 |
1030 |
10039 |
99822 |
999685 |
9997305 |
100013440 |
1000027552 |
REFERENCES:
Sloane, N. J. A. Sequences A002392/M0394, A228240,A228243, and A229197 in "The On-Line Encyclopedia of Integer Sequences."
Yee, A. J. "y-cruncher - A Multi-Threaded Pi-Program." http://www.numberworld.org/y-cruncher/.
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