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Natural Logarithm of 10 Digits


The numerical value of ln10 is given by

 ln10=2.302585092994045684...

(OEIS A002392). It was computed to 10^(11) decimal digits by S. Kondo on May 20, 2011 (Yee).

The Earls sequence (starting position of n copies of the digit n) for ln10 is given for n=1, 2, ... by 20, 111, 56, 9041, 4767, 674596, 24611354, 64653957, 131278082, ... (OEIS A228243).

ln10-constant primes occur at 1, 2, 40, 242, 842, 1541, 75067, ... decimal digits (OEIS A228240).

The starting positions of the first occurrence of n=0, 1, ... in the decimal expansion of ln10 (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 ln10 until all n-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 10^(10) decimal digits of ln10 (E. Weisstein, Aug. 27, 2013).

It is not known if ln10 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).

d\nOEIS1010010^310^410^510^610^710^810^910^(10)
0A00000021410396499799988710003569996731999878421000052045
1A000000058210391008810042999834099992451000080601000031412
2A00000021110610089943999271000879100051331000139391000008327
3A000000188110451010810016699932910001639100010112999996738
4A0000000910697310006100210998402999994899988853999996641
5A00000029103967996199375100138110000670100011829999956649
6A000000091079679828997699994891000118299972292999972639
7A000000011110100710061997821000327999962399989099999983529
8A0000001101061000998710063310018129998524100004534999974468
9A000000214961030100399982299968599973051000134401000027552

See also

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

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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

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