A065648 -id:A065648 - cp's OEIS frontend

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

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

A065649 Permutation of nonnegative integers based on Champernowne's constant 0.123456789101112131415...

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 21, 31, 41, 12, 51, 13, 61, 14, 71, 15, 81, 16, 91, 17, 101, 18, 111, 19, 22, 20, 32, 121, 42, 52, 62, 23, 72, 24, 82, 25, 92, 26, 102, 27, 112, 28, 122, 29, 33, 30, 43, 131, 53, 132, 63, 73, 83, 34, 93, 35, 103, 36, 113, 37, 123, 38

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Nov 09 2001

A261293(n) = a(a(n)). - Reinhard Zumkeller, Aug 14 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Klaus Strassburger, Illustration
Index entries for sequences that are permutations of the natural numbers

A033307, A065648, A065650.

Cf. A261279 (fixed points), A261293, A261333.

a065649 n = a065649_list !! n
a065649_list = zipWith (+)
               (map ((* 10) . subtract 1) a065648_list) (0 : a033307_list)
-- Reinhard Zumkeller, Aug 13 2015

a(n) = if n = 0 then 0 else 10*(A065648(n)-1) + A033307(n-1).

Offset and defining formula adjusted by Reinhard Zumkeller, Aug 13 2015

A065650 Inverse permutation to A065649.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 15, 17, 19, 21, 23, 25, 27, 29, 31, 12, 30, 37, 39, 41, 43, 45, 47, 49, 51, 13, 32, 50, 59, 61, 63, 65, 67, 69, 71, 14, 34, 52, 70, 81, 83, 85, 87, 89, 91, 16, 35, 54, 72, 90, 103, 105, 107, 109, 111, 18, 36, 56, 74, 92, 110, 125, 127

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Nov 09 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A033307, A065648, A065649, A261334.

Cf. A261279 (fixed points), A261294.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a065650 = fromJust . (`elemIndex` a065649_list)
-- Reinhard Zumkeller, Aug 13 2015

Offset changed by Reinhard Zumkeller, Aug 13 2015

A256100 In S = A007376 (read as a sequence) the digit S(n) appears a(n) times in the sequence S(1), ..., S(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 3, 4, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 3, 2, 4, 13, 5, 6, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 4, 3, 5, 14, 6, 14, 7, 8, 9, 4, 10, 4, 11, 4, 12, 4, 13, 4, 14, 4, 5, 4, 6, 15, 7, 15, 8, 15, 9, 10, 11, 5, 12, 5, 13, 5, 14, 5, 15, 5, 6

Offset: 1

Wolfdieter Lang, Apr 08 2015

The motivation to consider this sequence came from the proposal A256379 by Anthony Sand.
This sequence can also be read as an irregular triangle (array) in which a(n, k) is the number of appearances of the k-th digit of n in the digits of 1, ... ,n-1 and the first k digits of n. See the example for the head of this array. The row length is A055842(n), n >= 1.
This can also be described as the ordinal transform of A007376. - Franklin T. Adams-Watters, Oct 10 2015

			a(10) = 2 because A007376(10) = 1 and that sequence up to n=10 is 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, and 1 appears twice.
a(24) = 10 because A007376(24) = 1 and this is the tenth 1 in A007376 up to, and including, A007376(24).
Read as a tabf array a(n, k) with row length A055842(n) this begins:
   n\k  1  2  ...
   1:   1
   2:   1
   3:   1
   4:   1
   5:   1
   6:   1
   7:   1
   8:   1
   9:   1
  10:   2  1
  11:   3  4
  12:   5  2
  13:   6  2
  14:   7  2
  15:   8  2
  16:   9  2
  17:  10  2
  18:  11  2
  19:  12  2
  20:   3  2
  ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A007376, A065648.

a256100 n = a256100_list !! (n-1)
a256100_list = f a007376_list $ take 10 $ repeat 1 where
   f (d:ds) counts = y : f ds (xs ++ (y + 1) : ys) where
                           (xs, y:ys) = splitAt d counts
-- Reinhard Zumkeller, Aug 13 2015

lim = 120; s = Flatten[IntegerDigits /@ Range@ lim]; f[n_] := Block[{d = IntegerDigits /@ Take[s, n] // Flatten // FromDigits}, DigitCount[d][[If[ s[[n]] == 0, 10, s[[n]] ]] ] ]; Array[f, lim] (* Michael De Vlieger, Apr 08 2015, after Robert G. Wilson v at A007376 *)

a(n) gives the number of digits A007376(n) in the sequence starting with A007376(1) and ending with A007376(n).

A359663 a(1) = 1; for n > 1, a(n) is the smallest positive number which has not appeared that shares a factor with the sum of the first n terms of the Champernowne string starting from 1.

Original entry on oeis.org

1, 3, 2, 4, 5, 6, 7, 8, 9, 10, 12, 47, 14, 21, 15, 13, 11, 16, 18, 61, 20, 67, 73, 22, 24, 26, 25, 28, 30, 17, 27, 32, 33, 107, 109, 36, 19, 29, 34, 38, 127, 39, 35, 137, 40, 42, 44, 45, 48, 46, 49, 51, 43, 50, 54, 52, 57, 31, 55, 193, 56, 60, 23, 58, 62, 64, 63, 66, 65, 68, 70, 72, 69, 77, 75, 37

Offset: 1

Scott R. Shannon, Jan 10 2023

For the Champernowne string starting from 1 see A033307. In the first 100000 terms there are 45 fixed points: 1, 4, 5, 6, 7, ..., 252, 264, 319. It is plausible no more exist. The sequence is conjectured to be a permutation of the positive integers.

			a(3) = 2 as the sum of the first 3 terms of the Champernowne string is 1 + 2 + 3 = 6, and 2 is the smallest unused number that shares a factor with 6.
a(10) = 10 as the sum of the first 10 terms of the Champernowne string is 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 1 = 46, and 10 is the smallest unused number that shares a factor with 46.

Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Cf. A033307, A065648, A359114 (base-2), A027749.

cp's OEIS Frontend

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

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A065649 Permutation of nonnegative integers based on Champernowne's constant 0.123456789101112131415...

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A065650 Inverse permutation to A065649.

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A256100 In S = A007376 (read as a sequence) the digit S(n) appears a(n) times in the sequence S(1), ..., S(n).

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A359663 a(1) = 1; for n > 1, a(n) is the smallest positive number which has not appeared that shares a factor with the sum of the first n terms of the Champernowne string starting from 1.

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