Natural Logarithm of 2 Digits

The decimal expansion of the natural logarithm of 2 is given by

(OEIS A002162). It was computed to  decimal digits by S. Kondo on May 14, 2011 (Yee).

The Earls sequence (starting position of  copies of the digit ) for  is given for , 2, ... by 4, 419, 2114, 3929, 38451, 716837, 6180096, 10680693, 2539803904 (OEIS A228242).

-constant primes occur at 321, 466, 1271, 15690, 18872, 89973, ... decimal digits (OEIS A228226).

The starting positions of the first occurrence of , 1, ... in the decimal expansion of  are 9, 4, 22, 3, 5, 10, 1, 6, 8, ... (OEIS A100077).

Scanning the decimal expansion of  until all -digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 2, 98, 604, 1155, 46847, 175403, ... (OEIS A036901), which end at digits 22, 444, 7655, 98370, 1107795, 12983306, ... (OEIS A036905).

The digit string 0123456789 occurs starting at positions 3157027485, 8102152328, ... in the decimal digits of , and 9876543210 occurs starting at position 380113805, with no other occurrences in the first  digits (E. Weisstein, Aug. 20, 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 .

REFERENCES:

Sloane, N. J. A. Sequences A002162/M4074, A036901, A036905, A100077, A228226, and A228242 in "The On-Line Encyclopedia of Integer Sequences."

Yee, A. J. "y-cruncher - A Multi-Threaded Pi-Program." http://www.numberworld.org/y-cruncher/.