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A033307 Decimal expansion of Champernowne constant (or Mahler's number), formed by concatenating the positive integers. 73
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.

As n -> infinity, lim((A007908(n))/(prod(i=1,n, 10^floor(1+(log(i)/(log(10))))))) yields the Champernowne constant. - Alexander R. Povolotsky, May 29 2008, _Paolo Lava-, Jun 06 2008

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.

LINKS

Harry J. Smith, Table of n, a(n) for n = 0..20000

Bailey, D. H. and Crandall, R. E., Random Generators and Normal Numbers, Exper. Math. 11, 527-546, 2002.

Copeland, A. H. and Erdös, P., 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

Eric Weisstein's World of Mathematics, Champernowne constant

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.

EXAMPLE

0.12345678910111213141516171819202122232425262728293031323334353637383940\

4142434445464748495051525354555657585960616263646566676869707172737475767\

7787980...

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}] (* Robert G. Wilson v, Jul 23 2012 *)

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.

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

Sequence in context: A048379 A169930 A179295 * A007376 A189823 A001073

Adjacent sequences:  A033304 A033305 A033306 * A033308 A033309 A033310

KEYWORD

nonn,base,cons,easy

AUTHOR

Eric W. Weisstein

EXTENSIONS

Fixed my PARI program, had -n Harry J. Smith, May 19 2009

STATUS

approved

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Last modified April 20 16:20 EDT 2014. Contains 240806 sequences.