

A033307


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


108



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
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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
Named after David Gawen Champernowne, (July 09, 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 https://chessprogramming.wikispaces.com/David+Champernowne  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), 149166, 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
D. H. Bailey, and R. E. Crandall, Random Generators and Normal Numbers, Exper. Math. 11, 527546, 2002.
E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822829.
D. G. Champernowne, The Construction of Decimals Normal in the Scale of Ten, J. London Math. Soc., 8 (1933), 254260.
A. H. Copeland, and P. Erdös, Note on Normal Numbers, Bull. Amer. Math. Soc. 52, 857860, 1946.
Simon Plouffe, Champernowne constant, the natural integers concatenated
Simon Plouffe, Champernowne constant, the natural integers concatenated
Simon Plouffe, Generalized expansion of real constants
John K. Sikora, On the High Water Mark Convergents of Chamernowne's Constant in Base Ten.
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

0.12345678910111213141516171819202122232425262728293031323334353637383940414243...


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 July 4 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=(xd)*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.
Tables in which the nth 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


STATUS

approved



