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A033307
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Decimal expansion of Champernowne constant (or Mahler's number), formed by concatenating the positive integers.
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43
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| This number is known to be normal in base 10.
As n -> infinity, lim((A007908(n))/(prod(i=1,n, 10^floor(1+(log(i)/(log(10))))))) yields the Champernowne constant. - Alexander R. Povolotsky (pevnev(AT)juno.com), May 29 2008, Paolo Lava, Jun 06 2008
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REFERENCES
| Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.
E. B. Burger, Diophantine Olympics ..., Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
Champernowne, D. G. "The Construction of Decimals Normal in the Scale of Ten." J. London Math. Soc., 8 (1933), 254-260.
Copeland, A. H. and Erdos, P. "Note on Normal Numbers." Bull. Amer. Math. Soc. 52, 857-860, 1946.
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.
H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970, p. 172.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 364.
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LINKS
| Harry J. Smith, Table of n, a(n) for n=0,...,20000
S. Plouffe, Champernowne constant, the natural integers concatenated
S. Plouffe, Champernowne constant, the natural integers concatenated
S. Plouffe, Generalized expansion of real constants
Eric Weisstein's World of Mathematics, Champernowne constant
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FORMULA
| Formula for a(n) from David W. Cantrell, Feb 18, 2007: 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) = mod(floor(10^(mod(n + (10^i - 10)/9, i) - i + 1) ceiling((9n + 10^i - 1)/(9i) - 1)), 10). See also Mathematica code.
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EXAMPLE
| 0.12345678910111213141516171819202122232425262728293031323334353637383940\
4142434445464748495051525354555657585960616263646566676869707172737475767\
7787980...
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MATHEMATICA
| Flatten[IntegerDigits/@Range[0, 57]] (* Or *)
a[n_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = 9i*10^(i - 1) + l; i++ ]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + 10^(i - 1); If[p != 0, IntegerDigits[q][[p]], Mod[q - 1, 10]]]; Table[ a[n], {n, 0, 104}]
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]
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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)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 20 2009]
(Haskell)
import Data.Char (digitToInt)
a033307 n = a033307_list !! n
a033307_list = map digitToInt $ concatMap show [1..]
-- Reinhard Zumkeller, Mar 28 2011
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CROSSREFS
| See A030167 for the continued fraction expansion of this number.
A007376 is the same sequence but with a different interpretation.
Cf. A007908.
Sequence in context: A048379 A169930 A179295 * A007376 A189823 A001073
Adjacent sequences: A033304 A033305 A033306 * A033308 A033309 A033310
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KEYWORD
| nonn,cons,base
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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EXTENSIONS
| Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009
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