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%I #141 May 20 2023 15:13:09
%S 1,2,3,4,5,6,7,8,9,1,0,1,1,1,2,1,3,1,4,1,5,1,6,1,7,1,8,1,9,2,0,2,1,2,
%T 2,2,3,2,4,2,5,2,6,2,7,2,8,2,9,3,0,3,1,3,2,3,3,3,4,3,5,3,6,3,7,3,8,3,
%U 9,4,0,4,1,4,2,4,3,4,4,4,5,4,6,4,7,4,8,4,9,5,0,5,1,5,2,5,3,5,4,5,5,5,6,5
%N Decimal expansion of Champernowne constant (or Mahler's number), formed by concatenating the positive integers.
%C This number is known to be normal in base 10.
%C Named after David Gawen Champernowne (July 9, 1912 - August 19, 2000). - _Robert G. Wilson v_, Jun 29 2014
%D G. Harman, One hundred years of normal numbers, in M. A. Bennett et al., eds., Number Theory for the Millennium, II (Urbana, IL, 2000), 149-166, A K Peters, Natick, MA, 2002.
%D C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 364.
%D H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 172.
%H Harry J. Smith, <a href="/A033307/b033307.txt">Table of n, a(n) for n = 0..20000</a>
%H D. H. Bailey and R. E. Crandall, <a href="http://www.emis.de/journals/EM/expmath/volumes/11/11.4/pp527_546.pdf">Random Generators and Normal Numbers</a>, Exper. Math. 11, 527-546, 2002.
%H Maya Bar-Hillel and Willem A. Wagenaar, <a href="https://doi.org/10.1016/0196-8858(91)90029-I">The perception of randomness</a>, Advances in applied mathematics 12.4 (1991): 428-454. See page 428.
%H Edward B. Burger, <a href="http://www.maa.org/sites/default/files/pdf/upload_library/22/Chauvenet/Burger.pdf">Diophantine Olympics and World Champions: Polynomials and Primes Down Under</a>, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
%H Chess Programming Wiki, <a href="https://www.chessprogramming.org/David_Champernowne">David Champernowne</a> (as of Dec. 2019).
%H D. G. Champernowne, <a href="https://doi.org/10.1112/jlms/s1-8.4.254">The Construction of Decimals Normal in the Scale of Ten</a>, J. London Math. Soc., 8 (1933), 254-260.
%H Arthur H. Copeland and Paul Erdős, <a href="http://dx.doi.org/10.1090/S0002-9904-1946-08657-7">Note on Normal Numbers</a>, Bull. Amer. Math. Soc. 52, 857-860, 1946.
%H Peyman Nasehpour, <a href="https://arxiv.org/abs/1806.07560">A Simple Criterion for Irrationality of Some Real Numbers</a>, arXiv:1806.07560 [math.AC], 2018.
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap12.html">Champernowne constant, the natural integers concatenated</a>
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/champernowne.txt">Champernowne constant, the natural integers concatenated</a>
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/gendev/123456.html">Generalized expansion of real constants</a>
%H Paul Pollack and Joseph Vandehey, <a href="http://arxiv.org/abs/1405.6266">Besicovitch, Bisection, and the normality of 0.(1)(4)(9)(16)(25)...</a>, arXiv:1405.6266 [math.NT], 2014.
%H Paul Pollack and Joseph Vandehey, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.8.757">Besicovitch, Bisection, and the Normality of 0.(1)(4)(9)(16)(25)...</a>, The American Mathematical Monthly 122.8 (2015): 757-765.
%H John K. Sikora, <a href="http://arxiv.org/abs/1210.1263">On the High Water Mark Convergents of Champernowne's Constant in Base Ten</a>, arXiv:1210.1263 [math.NT], 2012.
%H John K. Sikora, <a href="http://arxiv.org/abs/1408.0261">Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases</a>, arXiv:1408.0261 [math.NT], 2014.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChampernowneConstant.html">Champernowne constant</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Champernowne_constant">Champernowne constant</a>.
%H Hector Zenil, N. Kiani and J. Tegner, <a href="https://arxiv.org/abs/1608.05972">Low Algorithmic Complexity Entropy-deceiving Graphs</a>, arXiv preprint arXiv:1608.05972 [cs.IT], 2016.
%F Let "index" i = ceiling( W(log(10)/10^(1/9) (n - 1/9))/log(10) + 1/9 ) where W denotes the principal branch of the Lambert W function. Then a(n) = (10^((n + (10^i - 10)/9) mod i - i + 1) * ceiling((9n + 10^i - 1)/(9i) - 1)) mod 10. See also Mathematica code. - David W. Cantrell, Feb 18 2007
%e 0.12345678910111213141516171819202122232425262728293031323334353637383940414243...
%t Flatten[IntegerDigits/@Range[0, 57]] (* Or *)
%t almostNatural[n_, b_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = (b - 1) i*b^(i - 1) + l; i++]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + b^(i - 1); If[p != 0, IntegerDigits[q, b][[p]], Mod[q - 1, b]]]; Array[ almostNatural[#, 10] &, 105] (* _Robert G. Wilson v_, Jul 23 2012 and modified Jul 04 2014 *)
%t intermediate[n_] := Ceiling[FullSimplify[ProductLog[Log[10]/10^(1/9) (n - 1/9)] / Log[10] + 1/9]]; champerDigit[n_] := Mod[Floor[10^(Mod[n + (10^intermediate[n] - 10)/9, intermediate[n]] - intermediate[n] + 1) Ceiling[(9n + 10^intermediate[n] - 1)/(9intermediate[n]) - 1]], 10]; (* David W. Cantrell, Feb 18 2007 *)
%o (PARI) { default(realprecision, 20080); x=0; y=1; d=10.0; e=1.0; n=0; while (y!=x, y=x; n++; if (n==d, d=d*10); e=e*d; x=x+n/e; ); d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b033307.txt", n, " ", d)); } \\ _Harry J. Smith_, Apr 20 2009
%o (Haskell)
%o a033307 n = a033307_list !! n
%o a033307_list = concatMap (map (read . return) . show) [1..] :: [Int]
%o -- _Reinhard Zumkeller_, Aug 27 2013, Mar 28 2011
%o (Magma) &cat[Reverse(IntegerToSequence(n)):n in[1..50]]; // _Jason Kimberley_, Dec 07 2012
%o (Scala) val numerStr = (1 to 100).map(Integer.toString(_)).toList.reduce(_ + _)
%o numerStr.split("").map(Integer.parseInt(_)).toList // _Alonso del Arte_, Nov 04 2019
%o (Python)
%o from itertools import count
%o def agen():
%o for k in count(1): yield from list(map(int, str(k)))
%o a = agen()
%o print([next(a) for i in range(104)]) # _Michael S. Branicky_, Sep 13 2021
%Y See A030167 for the continued fraction expansion of this number.
%Y A007376 is the same sequence but with a different interpretation.
%Y Cf. A007908, A000027, A001191 (concatenate squares).
%Y Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b = 2), A003137 and A054635 (b = 3), A030373 (b = 4), A031219 (b = 5), A030548 (b = 6), A030998 (b = 7), A031035 and A054634 (b = 8), A031076 (b = 9), A007376 and this sequence (b = 10). - _Jason Kimberley_, Dec 06 2012
%Y Cf. A065648.
%K nonn,base,cons,easy
%O 0,2
%A _Eric W. Weisstein_
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