A033307 Decimal expansion of Champernowne constant (or Mahler's number), formed by concatenating the positive integers.

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, 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, 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

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn
• base
• cons
• easy

Comments

This number is known to be normal in base 10.

Named after David Gawen Champernowne (July 9, 1912 - August 19, 2000). - Robert G. Wilson v, Jun 29 2014

Examples

			0.12345678910111213141516171819202122232425262728293031323334353637383940414243...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.9, p. 442.
• 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.
• C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 364.
• H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 172.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• D. H. Bailey and R. E. Crandall, Random Generators and Normal Numbers, Exper. Math. 11, 527-546, 2002.
• Maya Bar-Hillel and Willem A. Wagenaar, The perception of randomness, Advances in applied mathematics 12.4 (1991): 428-454. See page 428.
• Edward B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
• Chess Programming Wiki, David Champernowne (as of Dec. 2019).
• D. G. Champernowne, The Construction of Decimals Normal in the Scale of Ten, J. London Math. Soc., 8 (1933), 254-260.
• Arthur H. Copeland and Paul Erdős, Note on Normal Numbers, Bull. Amer. Math. Soc. 52, 857-860, 1946.
• Peyman Nasehpour, A Simple Criterion for Irrationality of Some Real Numbers, arXiv:1806.07560 [math.AC], 2018.
• Simon Plouffe, Champernowne constant, the natural integers concatenated.
• Simon Plouffe, Champernowne constant, the natural integers concatenated.
• Simon Plouffe, Generalized expansion of real constants.
• Paul Pollack and Joseph Vandehey, Besicovitch, Bisection, and the normality of 0.(1)(4)(9)(16)(25)..., arXiv:1405.6266 [math.NT], 2014.
• Paul Pollack and Joseph Vandehey, Besicovitch, Bisection, and the Normality of 0.(1)(4)(9)(16)(25)..., The American Mathematical Monthly 122.8 (2015): 757-765.
• John K. Sikora, On the High Water Mark Convergents of Champernowne's Constant in Base Ten, arXiv:1210.1263 [math.NT], 2012.
• John K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT], 2014.
• Eric Weisstein's World of Mathematics, Champernowne constant.
• Wikipedia, Champernowne constant.
• Hector Zenil, N. Kiani and J. Tegner, Low Algorithmic Complexity Entropy-deceiving Graphs, arXiv preprint arXiv:1608.05972 [cs.IT], 2016.
• Index entries for transcendental numbers

Crossrefs

See A030167 for the continued fraction expansion of this number.

A007376 is the same sequence but with a different interpretation.

Cf. A007908, A000027, A001191 (concatenate squares).

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

Cf. A065648.

Cf. A365237 (reciprocal).

Programs

Haskell

a033307 n = a033307_list !! n
a033307_list = concatMap (map (read . return) . show) [1..] :: [Int]
-- Reinhard Zumkeller, Aug 27 2013, Mar 28 2011

Magma

&cat[Reverse(IntegerToSequence(n)):n in[1..50]]; // Jason Kimberley, Dec 07 2012

Mathematica

Flatten[IntegerDigits/@Range[0, 57]] (* Or *)
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 *)
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 *)
First[RealDigits[ChampernowneNumber[], 10, 100]] (* Paolo Xausa, May 02 2024 *)

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

Python

from itertools import count
def agen():
    for k in count(1): yield from list(map(int, str(k)))
a = agen()
print([next(a) for i in range(104)]) # Michael S. Branicky, Sep 13 2021

Scala

val numerStr = (1 to 100).map(Integer.toString()).toList.reduce( + _)
numerStr.split("").map(Integer.parseInt()).toList // _Alonso del Arte, Nov 04 2019

Formula

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