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Search: id:A033307
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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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+0
40
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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; cons; graph; listen)
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OFFSET
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0,2
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COMMENT
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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
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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
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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
Eric Weisstein's World of Mathematics, Champernowne constant
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FORMULA
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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
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0.12345678910111213141516171819202122232425262728293031323334353637383940\
4142434445464748495051525354555657585960616263646566676869707172737475767\
7787980...
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MATHEMATICA
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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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PROGRAM
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(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]
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CROSSREFS
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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: A083116 A084044 A048379 this_sequence A007376 A001073 A076313
Adjacent sequences: A033304 A033305 A033306 this_sequence A033308 A033309 A033310
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KEYWORD
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nonn,cons
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AUTHOR
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Eric Weisstein (eric(AT)weisstein.com)
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EXTENSIONS
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Fixed my PARI program, had -n Harry J. Smith (hjsmithh(AT)sbcglobal.net), May 19 2009
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