TOPICS

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

Constant Digit Scanning, Constant Primes, Natural Logarithm of 10, Natural Logarithm of 10 Continued Fraction

Explore with Wolfram|Alpha

More things to try:

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/.

Cite this as:

Weisstein, Eric W. "Natural Logarithm of 10 Digits." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NaturalLogarithmof10Digits.html