This site is supported by donations to The OEIS Foundation.

 Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS". Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A033307 Decimal expansion of Champernowne constant (or Mahler's number), formed by concatenating the positive integers. 116
 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This number is known to be normal in base 10. Lim_{n->infinity}(A007908(n)/Product_{i=1..n} 10^floor(1+(log(i)/log(10)))) yields the Champernowne constant. - Alexander R. Povolotsky, May 29 2008, Paolo P. Lava, Jun 06 2008 Named after David Gawen Champernowne (July 9, 1912 - August 19, 2000), an English mathematician and economist who picked a hole in John Maynard Keynes's "General Theory of Employment, Interest and Money" and "built a chess computer" with Alan Turing, a longtime friend from the time that they were undergraduates together at King's College, Cambridge. See the Chess programming Wiki link. - Robert G. Wilson v, Jun 29 2014 REFERENCES 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. Bar-Hillel, Maya, and Willem A. Wagenaar. "The perception of randomness." Advances in applied mathematics 12.4 (1991): 428-454. See page 428. 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. E. 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. D. G. Champernowne, The Construction of Decimals Normal in the Scale of Ten, J. London Math. Soc., 8 (1933), 254-260. A. H. Copeland, and P. Erdős, Note on Normal Numbers, Bull. Amer. Math. Soc. 52, 857-860, 1946. 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, Joseph Vandehey, Besicovitch, Bisection, and the normality of 0.(1)(4)(9)(16)(25)..., arXiv:1405.6266 [math.NT], 2014. Paul Pollack, 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 Hector Zenil, N. Kiani, J. Tegner, Low Algorithmic Complexity Entropy-deceiving Graphs, arXiv preprint arXiv:1608.05972 [cs.IT], 2016. 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 EXAMPLE 0.12345678910111213141516171819202122232425262728293031323334353637383940414243... 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 *) i[n_] := Ceiling[FullSimplify[ProductLog[Log[10]/10^(1/9) (n - 1/9)] /Log[10] + 1/9]]; a[n_] := Mod[Floor[10^(Mod[n + (10^i[n] - 10)/9, i[n]] - i[n] + 1) Ceiling[(9n + 10^i[n] - 1)/(9i[n]) - 1]], 10]; (* David W. Cantrell, Feb 18 2007 *) PROG (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 (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 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. Sequence in context: A084044 A169930 A048379 * A007376 A189823 A001073 Adjacent sequences:  A033304 A033305 A033306 * A033308 A033309 A033310 KEYWORD nonn,base,cons,easy AUTHOR STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 18 16:24 EST 2017. Contains 296177 sequences.