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Champernowne Constant Digits

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The Champernowne constant has decimal expansion

C=0.1234567891011...

(OEIS A033307).

The Earls sequence (starting position of n copies of the digit n) for e is given for n=1, 2, ... by 1, 34, 56, 1222, 1555, 25554, 29998, 433330, 7988888882, 1101010101010, ... (OEIS A224896).

The starting positions of the first occurrence of n=0, 1, 2, ... in the decimal expansion of C (not including the initial 0 to the left of the decimal point) are 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 1, ... (OEIS A229186).

Scanning the decimal expansion of ln10 until all n-digit numbers have occurred, the last 1-, 2-, ... digit numbers appearing are 0, 00, 000, 0000, ..., which end at digits 11, 192, 2893, 38894, 488895, ... (OEIS A072290).

The digit sequence 0123456789 first occurs at positions 11234567799, 22345677908, 33456779017, 44567790126, 55677901235, 66779012344, ... (OEIS A000000) and 9876543210 at positions 7777777779, 9876543212, 19987654323, 30998765434, 42099876545, 53209987656, 64320998767, ... (OEIS A000000; E. Weisstein, Jul. 26, 2013).

C-constant primes occur for 10, 14, 24, 235, 2804, 4347, 37735, ... (OEIS A071620) digits.

It is known that the Champernowne constant is normal in base 10 (Champernowne 1933, Bailey and Crandall 2002), though the following table giving the counts of digits in the first 10^n terms shows non-normal behavior up to at least 10^9 due to an excess of 1s and surfeit of 0s when cutting the digit string off at locations such as ...123456787123456788.

d\nOEIS1010010^310^410^510^610^710^810^9
0A0000000566747864283528884151923456896021948
1A000000216177185819753179810158256214234568130589850
2A0000001161771636111119453999526010345679100589849
3A0000001161488588642945399951611023456896589849
4A00000011677858864294539995160934567996089849
5A00000011177858864293723982462934567996029849
6A0000001577858864293538895160934567996022849
7A0000001567833864293538894462934567996022049
8A0000001567747864288718891462933333396021959
9A0000001567747864283528884160923456896021949

See also

Champernowne Constant, Champernowne Constant Continued Fraction, Constant Digit Scanning, Constant Primes, Earls Sequence, Smarandache Number

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References

Champernowne, D. G. "The Construction of Decimals Normal in the Scale of Ten." J. London Math. Soc. 8, 1933.Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.Sloane, N. J. A. Sequences A071620, A072290, A224896, and A229186 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Champernowne Constant Digits." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ChampernowneConstantDigits.html

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