A365238 -id:A365238 - cp's OEIS frontend

A066716 Decimal expansion of the binary Champernowne constant 0.862240125868... whose binary expansion is the concatenation of 1, 2, 3, ... written in binary.

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

Offset: 0

Robert G. Wilson v, Jan 14 2002

A theorem of Copeland & Erdős proves that this constant is 2-normal. - Charles R Greathouse IV, Feb 06 2015

This constant is transcendental. Note that this result is nontrivial: it is not a corollary of the result of Masaaki Amou saying that the base-b Champernowne constant has irrationality measure b, because the Thue-Siegel-Roth theorem only guarantees that a number with irrationality measure greater than 2 is transcendental. However, it is already stated in Masaaki Amou's paper that K. Mahler proved that the base-b Champernowne constant is transcendental for all b. - Jianing Song, Sep 27 2023

			0.8622401258680545715577902832493945785657647427682990945160712145573067405905...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Masaaki Amou, Approximation to certain transcendental decimal fractions by algebraic numbers, J. Number Theory, 37 (2) (1991), pp. 231-241.
A. H. Copeland and P. Erdős, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), pp. 857-860.
Eric E. Weisstein, Binary Champernowne Constant.

Cf. A030302 (binary digits), A030190 (same with initial 0), A030303 (indices of 1's), A007088, A047778 (concatenate binary 1..n).
Cf. A066717 (continued fraction), A365238 (reciprocal).
Cf. A100125 (Sum n/2^(n^2)).
Cf. A033307.

a = {}; Do[a = Append[a, IntegerDigits[n, 2]], {n, 1, 100} ]; RealDigits[ N[ FromDigits[ {Flatten[a], 0}, 2], 100]]
First[RealDigits[ChampernowneNumber[2], 10, 100]] (* Paolo Xausa, Jun 12 2024 *)

my(s=0.); forstep(n=default(realprecision),1,-1,s=(s+n)>>#binary(n)); s \\ Charles R Greathouse IV, Feb 06 2015, corrected by M. F. Hasler, Mar 22 2017

s=0;sum(n=1,31,n*.5^s+=logint(n,2)+1) \\ Accurate to 0.5^s. The sum up to n=31 is enough for standard precision of 38 digits. - M. F. Hasler, Mar 22 2017

The "binary" Champernowne constant is the number whose base-2 expansion is the concatenation of the binary representations of the integers, 0.(1)(10)(11)(100)(101)(110)(111)(1000)..., cf. A030302.

Leading zero removed, offset adjusted, and keyword:cons added by R. J. Mathar, Mar 04 2010

Name edited by M. F. Hasler, Oct 26 2019

A365237 Decimal expansion of 1/A033307 (decimal Champernowne constant).

Original entry on oeis.org

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

Offset: 1

Kelvin Voskuijl, Aug 27 2023

Appears to coincide with A090903 from the 10th digit onwards.

			8.10000006707603361330731967383416787753583647347857972252510...

Cf. A031310, A033307.
Cf. A030167 (continued fraction).
Cf. A365238 (reciprocal of binary Champernowne constant).

RealDigits[1/ChampernowneNumber[10] , 10, 120][[1]]

cp's OEIS Frontend

A066716 Decimal expansion of the binary Champernowne constant 0.862240125868... whose binary expansion is the concatenation of 1, 2, 3, ... written in binary.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI