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 .