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Natural Logarithm of 2 Digits


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

 ln2=0.69314718055994530941...

(OEIS A002162). It was computed to 10^(11) decimal digits by S. Kondo on May 14, 2011 (Yee).

The Earls sequence (starting position of n copies of the digit n) for ln2 is given for n=1, 2, ... by 4, 419, 2114, 3929, 38451, 716837, 6180096, 10680693, 2539803904 (OEIS A228242).

ln2-constant primes occur at 321, 466, 1271, 15690, 18872, 89973, ... decimal digits (OEIS A228226).

The starting positions of the first occurrence of n=0, 1, ... in the decimal expansion of ln2 are 9, 4, 22, 3, 5, 10, 1, 6, 8, ... (OEIS A100077).

Scanning the decimal expansion of ln2 until all n-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 ln2, and 9876543210 occurs starting at position 380113805, with no other occurrences in the first 10^(10) digits (E. Weisstein, Aug. 20, 2013).

It is not known if ln2 is normal, but the following table giving the counts of digits in the first 10^n terms shows that the decimal digits are very uniformly distributed up to at least 10^(10).


See also

Constant Digit Scanning, Constant Primes, Natural Logarithm of 2, Natural Logarithm of 2 Continued Fraction

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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/.

Cite this as:

Weisstein, Eric W. "Natural Logarithm of 2 Digits." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NaturalLogarithmof2Digits.html

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