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e Digits


The constant e with decimal expansion

 e=2.718281828459045235360287471352662497757...

(OEIS A001113) can be computed to 10^9 digits of precision in 10 CPU-minutes on modern hardware.

e was computed to 1.7×10^9 digits by P. Demichel, and the first 1.25×10^9 have been verified by X. Gourdon on Nov. 21, 1999 (Plouffe). e was computed to 10^(12) decimal digits by S. Kondo on Jul. 5, 2010 (Yee).

The Earls sequence (starting position of n copies of the digit n) for e is given for n=1, 2, ... by 2, 252, 1361, 11806, 210482, 9030286, 3548262, 141850388, 1290227011, ... (OEIS A224828).

The starting positions of the first occurrence of n in the decimal expansion of e (including the initial 2 and counting it as the first digit) are 14, 3, 1, 18, 11, 12, 21, 2, ... (OEIS A088576).

Scanning the decimal expansion of e until all n-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 6, 12, 548, 1769, 92994, ... (OEIS A036900), which end at digits 21, 372, 8092, 102128, ... (OEIS A036904).

The digit sequence 0123456789 does not occur in the first 10^(10) digits of e, but 9876543210 does, starting at position 6001160363 (E. Weisstein, Jul. 22, 2013).

e-constant primes (i.e., e-primes) occur at 1, 3, 7, 85, 1781, 2780, 112280, 155025, ... (OEIS A64118) decimal digits.

It is not known if e 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)
0A0000000510097498859942599867899991381000044251000024802
1A00000026969891026410013210005771000443899982926999989229
2A000000212971004985599845999156999887699999168999997938
3A00000008109100810035100228100171610005176100002498999982936
4A00000011110098210039100389100030799982851000189221000026506
5A00000001385992100341000879999039998042100003884999967300
6A000000012991079101831004799988691000015899987241999931170
7A00000011699100898759991010008139998342999975361000013049
8A00000047103996996799814999703100003361000053481000074277
9A0000000101129689863996911000278999720999998052999992793

See also

Constant Digit Scanning, Constant Primes, e, e-Prime, Earls Sequence

Explore with Wolfram|Alpha

References

Sloane, N. J. A. Sequences A001113/M1727, A036900, A036904, A064118, A088576, and A224828 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. "e Digits." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/eDigits.html

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