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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\n OEIS 10 100 10^3 10^4 10^5 10^6 10^7 10^8 10^9 10^(10)
0 A000000 0 5 100 974 9885 99425 998678 9999138 100004425 1000024802
1 A000000 2 6 96 989 10264 100132 1000577 10004438 99982926 999989229
2 A000000 2 12 97 1004 9855 99845 999156 9998876 99999168 999997938
3 A000000 0 8 109 1008 10035 100228 1001716 10005176 100002498 999982936
4 A000000 1 11 100 982 10039 100389 1000307 9998285 100018922 1000026506
5 A000000 0 13 85 992 10034 100087 999903 9998042 100003884 999967300
6 A000000 0 12 99 1079 10183 100479 998869 10000158 99987241 999931170
7 A000000 1 16 99 1008 9875 99910 1000813 9998342 99997536 1000013049
8 A000000 4 7 103 996 9967 99814 999703 10000336 100005348 1000074277
9 A000000 0 10 112 968 9863 99691 1000278 9997209 99998052 999992793

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/.

Referenced on Wolfram|Alpha

e Digits

Cite this as:

Weisstein, Eric W. "e Digits." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/eDigits.html

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