A066716 -id:A066716 - cp's OEIS frontend

A378331 Decimal expansion of the base 7 Champernowne constant.

1, 9, 4, 4, 3, 5, 5, 3, 5, 0, 8, 6, 2, 4, 0, 5, 2, 1, 4, 7, 5, 8, 4, 0, 0, 9, 3, 0, 8, 2, 9, 0, 8, 5, 7, 6, 4, 5, 2, 9, 3, 2, 9, 7, 1, 0, 5, 0, 4, 2, 2, 1, 1, 2, 4, 7, 9, 5, 8, 8, 5, 3, 1, 2, 3, 3, 6, 7, 9, 0, 8, 8, 7, 3, 9, 4, 0, 3, 5, 6, 6, 3, 9, 7, 0, 8, 5

Offset: 0

Joshua Searle, Nov 25 2024

This constant is formed by the concatenation of the natural numbers in base 7 and then converted into base 10.

This constant is 7-normal.

			0.194435535086240521475840093082908576452932971050422112479588531233679088...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307
(base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[7], 10, 100]]

A378332 Decimal expansion of the base 8 Champernowne constant.

Original entry on oeis.org

1, 6, 3, 2, 6, 4, 8, 1, 2, 1, 0, 5, 2, 1, 6, 7, 9, 7, 3, 6, 7, 0, 9, 4, 9, 8, 6, 1, 4, 2, 6, 0, 5, 1, 9, 0, 2, 2, 4, 2, 3, 7, 8, 4, 3, 2, 8, 5, 4, 6, 2, 3, 3, 3, 0, 8, 1, 3, 8, 0, 7, 0, 0, 4, 2, 8, 3, 1, 9, 4, 7, 5, 9, 3, 8, 5, 2, 3, 5, 5, 7, 5, 7, 1, 1, 7, 6

Offset: 0

Joshua Searle, Nov 25 2024

This constant is formed by the concatenation of the natural numbers in base 8 and then converted into base 10.

This constant is 8-normal.

			0.163264812105216797367094986142605190224237843285462333081380700428319475...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[8], 10, 100]]

A378333 Decimal expansion of the base 9 Champernowne constant.

Original entry on oeis.org

1, 4, 0, 6, 2, 4, 9, 7, 6, 1, 1, 9, 6, 9, 6, 7, 8, 2, 4, 7, 9, 6, 6, 9, 0, 0, 8, 9, 3, 5, 6, 6, 3, 1, 8, 3, 2, 6, 5, 4, 5, 7, 0, 8, 3, 2, 4, 6, 8, 2, 8, 4, 8, 6, 6, 5, 7, 5, 5, 5, 1, 7, 1, 2, 7, 5, 4, 1, 4, 9, 1, 4, 8, 7, 8, 1, 8, 5, 4, 9, 5, 2, 4, 3, 6, 4, 4

Offset: 0

Joshua Searle, Nov 25 2024

This constant is formed by the concatenation of the natural numbers in base 9 and then converted into base 10.

This constant is 9-normal.

			0.140624976119696782479669008935663183265457083246828486657555171275414914...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[9], 10, 100]]

A353962 Square array read by descending antidiagonals: The n-th row gives the decimal expansion of the base-n Champernowne constant.

Original entry on oeis.org

8, 6, 5, 2, 9, 4, 2, 8, 2, 3, 4, 9, 6, 1, 2, 0, 5, 1, 0, 3, 1, 1, 8, 1, 7, 9, 9, 1, 2, 1, 1, 3, 8, 4, 6, 1, 5, 6, 1, 6, 6, 4, 3, 4, 1, 8, 7, 1, 1, 2, 3, 2, 0, 2, 1, 6, 5, 1, 1, 6, 5, 6, 6, 3, 0, 0, 8, 3, 1, 1, 8, 5, 4, 2, 4, 9, 9, 0, 0, 8, 1, 1, 5, 3, 8, 4, 5, 9, 9, 9, 0

Offset: 2

Davis Smith, May 12 2022

The base-n Champernowne constant (C_n) is normal in base n. A(n,k) is the (k+1)-th decimal digit of the fractional part of C_n.

			The square array A(n,k) begins:
  n/k | 0  1  2  3  4  5  6  7  8  9 10 11 ...
  ----+---------------------------------------
   2  | 8  6  2  2  4  0  1  2  5  8  6  8 ...
   3  | 5  9  8  9  5  8  1  6  7  5  3  8 ...
   4  | 4  2  6  1  1  1  1  1  1  1  1  1 ...
   5  | 3  1  0  7  3  6  1  1  1  1  1  1 ...
   6  | 2  3  9  8  6  2  6  8  5  8  1  5 ...
   7  | 1  9  4  4  3  5  5  3  5  0  8  6 ...
   8  | 1  6  3  2  6  4  8  1  2  1  0  5 ...
   9  | 1  4  0  6  2  4  9  7  6  1  1  9 ...
  10  | 1  2  3  4  5  6  7  8  9  1  0  1 ...
  ...

Verónica Becher and Santiago Figueira, An example of a computable absolutely normal number, Theoretical Computer Science, 270 (2002), 947-958.
Arthur H. Copeland and Paul Erdős, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), 857-860.
Davar Khoshnevisan, Normal Numbers are Normal, Clay Mathematics Institute Annual Report, 2006, 15-31.
Ivan Niven and H.S. Zuckerman, On The Definition of Normal Numbers, Pacific J. Math., 1 (1951), 103-109.
Davis Smith, A Sufficient Condition For Normalcy.

Rows: A066716 (n=2), A077771 (n=3), A033307 (n=10).

Cf. A063945.

A[n_,k_]:=Mod[Floor[ChampernowneNumber[n]10^(k + 1)] ,10]; Flatten[Table[Reverse[Table[A[n-k,k],{k,0,n-2}]],{n,2,14}]] (* Stefano Spezia, May 13 2022 *)

A(n,k) = floor(C_n*10^(k+1)) mod 10 where C_n (the base-n Champernowne constant) = Sum_{i>=1} i/(n^(i + Sum_{k=1..i-1} floor(log_n(k+1)))).

A100125 Decimal expansion of Sum_{n>0} n/(2^(n^2)).

Original entry on oeis.org

6, 3, 0, 9, 2, 0, 5, 5, 9, 2, 5, 5, 1, 8, 5, 8, 6, 4, 7, 7, 8, 3, 2, 4, 0, 0, 3, 9, 0, 7, 9, 4, 3, 3, 7, 0, 0, 9, 2, 1, 5, 1, 4, 2, 9, 9, 2, 1, 7, 8, 7, 9, 8, 6, 8, 0, 6, 4, 4, 4, 2, 4, 8, 9, 9, 9, 8, 9, 8, 0, 8, 1, 0, 7, 8, 3, 8, 1, 7, 7, 3, 4, 7, 3, 8, 8, 2, 0, 0, 1, 9, 2, 0, 6, 4, 4, 4, 5, 2, 1

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Nov 11 2004

This number is obviously 2-dense, but not 2-normal: any finite binary string s representing the value N will appear in its digits, not later than those added by the term N/2^(N^2), but nonzero digits have density zero since the gap between those added by subsequent terms is increasing much faster (~ n) than the maximal possible number of new nonzero digits (~ log_2(n)). - M. F. Hasler, Mar 22 2017

			0.6309205592551858647783240039079433700921514299217879868...

David H. Bailey and Richard E. Crandall, Random Generators and Normal Numbers, page 27.

Cf. A066716: binary Champernowne constant.

RealDigits[N[Sum[n/(2^(n^2)), {n, 4!}], 100]][[1]] (* Arkadiusz Wesolowski, Sep 29 2011 *)

default(realprecision,100);sum(n=1,100,n/(2^(n^2)),0.) \\ Typo corrected. sum(n=1,100,n*1.>>(n^2)) is 25 x faster for 1000 digits. - M. F. Hasler, Mar 22 2017

Offset corrected by Arkadiusz Wesolowski, Sep 29 2011

A180443 Decimal expansion of constant defined in A030315.

Original entry on oeis.org

5, 6, 8, 8, 7, 9, 9, 3, 7, 0, 6, 5, 9, 7, 2, 7, 1, 4, 2, 2, 1, 1, 0, 4, 8, 5, 8, 3, 7, 5, 3, 0, 2, 7, 1, 0, 7, 1, 7, 1, 1, 7, 6, 2, 8, 6, 1, 5, 8, 5, 0, 4, 5, 2, 7, 4, 1, 9, 6, 4, 3, 9, 2, 7, 2, 1, 3, 4, 6, 6, 2, 9, 7, 0, 4, 7, 4, 1, 7, 7, 0, 9, 7, 8, 9, 8, 0

Offset: 0

Jonathan Vos Post, Sep 05 2010

This constant is 2-normal. - Charles R Greathouse IV, Feb 06 2015

			0.56887993706597271422110485837530271071711762861585045274...

Paolo Xausa, Table of n, a(n) for n = 0..10000

Cf. A030190, A030315, A066716, A180433.

First[RealDigits[1 - ChampernowneNumber[2]/2, 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

Equals 1 - A066716 / 2. - Amiram Eldar, May 22 2023

a(30) onward corrected by Sean A. Irvine, Jun 09 2024

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A378331 Decimal expansion of the base 7 Champernowne constant.

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A378332 Decimal expansion of the base 8 Champernowne constant.

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A378333 Decimal expansion of the base 9 Champernowne constant.

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A353962 Square array read by descending antidiagonals: The n-th row gives the decimal expansion of the base-n Champernowne constant.

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A100125 Decimal expansion of Sum_{n>0} n/(2^(n^2)).

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A180443 Decimal expansion of constant defined in A030315.

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