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A066716 Decimal expansion of the binary Champernowne constant 0.862240125868... whose binary expansion is the concatenation of 1, 2, 3, ... written in binary. 10
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
A theorem of Copeland & Erdős proves that this constant is 2-normal. - Charles R Greathouse IV, Feb 06 2015
This is 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
LINKS
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.
FORMULA
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.
EXAMPLE
0.8622401258680545715577902832493945785657647427682990945160712145573067405905...
MATHEMATICA
a = {}; Do[a = Append[a, IntegerDigits[n, 2]], {n, 1, 100} ]; RealDigits[ N[ FromDigits[ {Flatten[a], 0}, 2], 100]]
PROG
(PARI) 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
(PARI) 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
CROSSREFS
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.
Sequence in context: A272430 A021120 A021541 * A339801 A360938 A272006
KEYWORD
cons,nonn,base
AUTHOR
Robert G. Wilson v, Jan 14 2002
EXTENSIONS
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
STATUS
approved

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)