A030167 -id:A030167 - cp's OEIS frontend

A038705 Position of the incrementally largest term in continued fraction for Champernowne constant (A030167).

1, 2, 3, 5, 19, 41, 163, 527, 1709, 4839, 13523, 34063

Offset: 1

Hans Havermann, May 01 2000

H. Havermann, Number of Digits in Some Champernowne-Continued-Fraction Terms
J. K. Sikora, Number of Digits in the CFE of Champernowne's Constant to the Coefficient Before HWM #12
J. K. Sikora, On the High Water Mark Convergents of Champernowne's Constant in Base Ten, arXiv:1210.1263 [math.NT]
J. K. Sikora, Champernowne's Constant CFE Coefficients to HWM #12 (789 MB unzipped)
J. K. Sikora, Champernowne's Constant CFE Calculation Program Source
Eric Weisstein's World of Mathematics, Champernowne Constant Continued Fraction

Cf. A030167, A038704, A038706.
See A143533 for another version.
Cf. A143532 (number of decimal digits in n-th term of c.f.).
Cf. A143534 (number of decimal digits in a(n)-th term of c.f.).

a(10) = 13523 (from Mark Sofroniou) contributed by Eric W. Weisstein, Sep 04 2008.
Edited by N. J. A. Sloane, Apr 03 2010.
a(11) = 34063 contributed by John K. Sikora, Aug 24 2012.

A038706 Position of the first occurrence of n in continued fraction for Champernowne constant (A030167).

Original entry on oeis.org

4, 28, 13, 9, 93, 20, 31, 2, 3, 339, 71, 126, 107, 48, 18, 292, 327, 133, 611, 342, 23, 202, 441, 164, 512, 235, 957, 349, 528, 77, 179, 591, 307, 381, 2011, 245, 2435, 187, 1870, 117, 1359, 1071, 762, 4199, 4309, 2308, 4294, 1012, 1501, 3490, 2249, 283

Offset: 1

Hans Havermann, May 01 2000

Only 187 terms of the sequence are known. The position of 188 is > 82328. - Eric W. Weisstein, Jun 29 2013

John K. Sikora and Eric W. Weisstein, Table of n, a(n) for n = 1..187 (terms n = 1..112 from John K. Sikora)

Cf. A030167, A038704, A038705.

A038704 Incrementally largest term in continued fraction for Champernowne constant (A030167).

Original entry on oeis.org

8, 9, 149083

Offset: 1

Hans Havermann, May 01 2000

The next term (a(4)) has 166 digits; a(5) has 2504 digits. - Harvey P. Dale, Sep 18 2016

H. Havermann, Number of Digits in Some Champernowne-Continued-Fraction Terms

Cf. A030167, A038705, A038706.

DeleteDuplicates[Block[{$MaxExtraPrecision=10000},ContinuedFraction[ ChampernowneNumber[],50]],GreaterEqual] (* Harvey P. Dale, Apr 02 2023 *)

The next term has 166 digits.

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

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein

This number is known to be normal in base 10.

Named after David Gawen Champernowne (July 9, 1912 - August 19, 2000). - Robert G. Wilson v, Jun 29 2014

			0.12345678910111213141516171819202122232425262728293031323334353637383940414243...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.9, p. 442.
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.
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.

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.
Maya Bar-Hillel and Willem A. Wagenaar, The perception of randomness, Advances in applied mathematics 12.4 (1991): 428-454. See page 428.
Edward 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 (as of Dec. 2019).
D. G. Champernowne, The Construction of Decimals Normal in the Scale of Ten, J. London Math. Soc., 8 (1933), 254-260.
Arthur H. Copeland and Paul Erdős, Note on Normal Numbers, Bull. Amer. Math. Soc. 52, 857-860, 1946.
Peyman Nasehpour, A Simple Criterion for Irrationality of Some Real Numbers, arXiv:1806.07560 [math.AC], 2018.
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 and Joseph Vandehey, Besicovitch, Bisection, and the normality of 0.(1)(4)(9)(16)(25)..., arXiv:1405.6266 [math.NT], 2014.
Paul Pollack and 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.
Wikipedia, Champernowne constant.
Hector Zenil, N. Kiani and J. Tegner, Low Algorithmic Complexity Entropy-deceiving Graphs, arXiv preprint arXiv:1608.05972 [cs.IT], 2016.
Index entries for transcendental numbers

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.
Cf. A365237 (reciprocal).

a033307 n = a033307_list !! n
a033307_list = concatMap (map (read . return) . show) [1..] :: [Int]
-- Reinhard Zumkeller, Aug 27 2013, Mar 28 2011

&cat[Reverse(IntegerToSequence(n)):n in[1..50]]; // Jason Kimberley, Dec 07 2012

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 *)
intermediate[n_] := Ceiling[FullSimplify[ProductLog[Log[10]/10^(1/9) (n - 1/9)] / Log[10] + 1/9]]; champerDigit[n_] := Mod[Floor[10^(Mod[n + (10^intermediate[n] - 10)/9, intermediate[n]] - intermediate[n] + 1) Ceiling[(9n + 10^intermediate[n] - 1)/(9intermediate[n]) - 1]], 10]; (* David W. Cantrell, Feb 18 2007 *)
First[RealDigits[ChampernowneNumber[], 10, 100]] (* Paolo Xausa, May 02 2024 *)

{ 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

from itertools import count
def agen():
    for k in count(1): yield from list(map(int, str(k)))
a = agen()
print([next(a) for i in range(104)]) # Michael S. Branicky, Sep 13 2021

val numerStr = (1 to 100).map(Integer.toString()).toList.reduce( + _)
numerStr.split("").map(Integer.parseInt()).toList // _Alonso del Arte, Nov 04 2019

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

A378345 Continued fraction expansion of the base 4 Champernowne constant.

Original entry on oeis.org

0, 2, 2, 1, 7, 1, 1, 2, 1, 1, 1, 1, 6806293849, 1, 33, 157, 1, 2, 1, 3, 1, 1, 2345427263108642344323518197756649380964709224412095403124301722165, 2, 2, 1, 1, 1, 3, 1, 1, 6, 2, 7, 11, 1, 1, 7, 12, 1, 1, 1, 126, 3, 13, 1, 13, 4, 33, 3, 1, 1, 1, 3, 2, 4, 1, 9, 2

Offset: 0

Joshua Searle, Dec 13 2024

OeisWiki, Champernowne constant

Cf. A030373 (base 4 expansion), A378328 (decimal expansion).

Other continued fractions: A066717, A077772, A378346, A378347, A378348, A378349, A378350, A030167.

ContinuedFraction[ChampernowneNumber[4], 100]

A378346 Continued fraction expansion of the base 5 Champernowne constant.

Original entry on oeis.org

0, 3, 4, 1, 1, 2, 2, 18, 1, 20, 1302701925685142513155, 3, 5, 6, 1, 1, 1, 1, 1, 1, 2, 13, 5, 2, 1, 22, 1, 1

Offset: 0

Joshua Searle, Dec 13 2024

The next term a(28) is approximately equal to 2.83 * 10^173.

OeisWiki, Champernowne constant

Cf. A031219 (base 5 expansion), A378329 (decimal exapansion).

Other continued fractions: A066717, A077772, A378345, A378347, A378348, A378349, A378350, A030167.

ContinuedFraction[ChampernowneNumber[5], 100]

A378347 Continued fraction expansion of the base 6 Champernowne constant.

Original entry on oeis.org

0, 4, 5, 1, 10, 1, 4, 3, 9, 1, 2, 2, 1, 1, 699745284439054751106354294914368414245, 2, 5, 1, 20, 22, 2, 2, 1, 10, 3, 1, 2, 2, 2, 1, 1, 2, 1, 1

Offset: 0

Joshua Searle, Dec 13 2024

The next term a(34) is approximately equal to 1.21 * 10^364.

OeisWiki, Champernowne constant

Cf. A030548 (base 6 expansion), A378330 (decimal expansion).

Other continued fractions: A066717, A077772, A378345, A378346, A378348, A378349, A378350, A030167.

ContinuedFraction[ChampernowneNumber[6], 100]

A378348 Continued fraction expansion of the base 7 Champernowne constant.

Original entry on oeis.org

0, 5, 6, 1, 85, 1, 2, 1, 11, 1, 3, 2, 1, 5, 1, 2, 8697444597678755989498288581049684565698396369776180853037564, 1, 4, 2, 8, 6, 1, 2, 11, 1, 11, 1, 9, 2, 11, 1, 13, 2, 3, 10

Offset: 0

Joshua Searle, Dec 14 2024

The next term a(36) is approximately equal to 4.24*10^662.

OeisWiki, Champernowne constant

Cf. A030998 (base 7 expansion), A378331 (decimal expansion).

Other continued fractions: A066717, A077772, A378345, A378346, A378347, A378349, A378350, A030167.

ContinuedFraction[ChampernowneNumber[7], 100]

A378349 Continued fraction expansion of the base 8 Champernowne constant.

Original entry on oeis.org

0, 6, 7, 1, 842, 5, 11, 2, 1, 4, 1, 12, 1217611913245203113561611289624720261608646275831638269345353220034950193075766082779756144, 39, 1, 13, 19, 1, 1, 2, 1, 6, 1, 4, 9, 1, 2, 1, 3, 2, 1, 223, 2, 1

Offset: 0

Joshua Searle, Dec 14 2024

The next term a(34) is approximately equal to 5.28 * 10^1099.

OeisWiki, Champernowne constant

Cf. A054634 (base 8 expansion), A378332 (decimal expansion).

Other continued fractions: A066717, A077772, A378345, A378346, A378347, A378348, A378350, A030167.

ContinuedFraction[ChampernowneNumber[8], 100]

A378350 Continued fraction expansion of the base 9 Champernowne constant.

Original entry on oeis.org

0, 7, 8, 1, 10222, 1, 1, 1, 1, 1, 12, 1, 1, 1, 145, 1, 13127841267973253934598674824559230051317913195904874825561053745645554655306632773083671838234108227370808367172269493508107, 1, 7, 3, 1, 1, 1, 2, 2, 15, 3, 2, 1, 3, 2, 1, 1, 7, 4, 1, 4, 1, 1, 3, 3, 1, 1

Offset: 0

Joshua Searle, Dec 14 2024

OeisWiki, Champernowne constant

Cf. A031076 (base 9 expansion), A378333 (decimal expansion).

Other continued fractions: A066717, A077772, A378345, A378346, A378347, A378348, A378349, A030167.

ContinuedFraction[ChampernowneNumber[9], 100]

cp's OEIS Frontend

A038705 Position of the incrementally largest term in continued fraction for Champernowne constant (A030167).

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A038706 Position of the first occurrence of n in continued fraction for Champernowne constant (A030167).

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A038704 Incrementally largest term in continued fraction for Champernowne constant (A030167).

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

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A378345 Continued fraction expansion of the base 4 Champernowne constant.

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A378346 Continued fraction expansion of the base 5 Champernowne constant.

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A378347 Continued fraction expansion of the base 6 Champernowne constant.

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A378348 Continued fraction expansion of the base 7 Champernowne constant.

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A378349 Continued fraction expansion of the base 8 Champernowne constant.

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