A030998 -id:A030998 - cp's OEIS frontend

A030999 Position of n-th 0 in A030998.

0, 8, 22, 36, 50, 64, 78, 92, 93, 95, 98, 101, 104, 107, 110, 114, 135, 156, 177, 198, 219, 239, 240, 242, 245, 248, 251, 254, 257, 261, 282, 303, 324, 345, 366, 386, 387, 389, 392, 395, 398, 401, 404, 408, 429, 450, 471, 492, 513

Offset: 1

Clark Kimberling

Harvey P. Dale, Table of n, a(n) for n = 1..1001

Rest[Flatten[Position[Flatten[IntegerDigits[#,7]&/@Range[0,200]], 0]]]-1 (* Harvey P. Dale, Oct 10 2011 *)

Missing a(1)=0 inserted by Sean A. Irvine, Apr 10 2020

A031000 Position of n-th 1 in A030998.

Original entry on oeis.org

1, 7, 9, 10, 11, 13, 15, 17, 19, 24, 38, 52, 66, 80, 91, 94, 96, 97, 100, 103, 106, 109, 112, 113, 115, 116, 117, 118, 119, 121, 122, 124, 125, 127, 128, 130, 131, 133, 136, 138, 139, 142, 145, 148, 151, 154, 157, 159, 160, 163, 166

Offset: 1

Clark Kimberling

nonn

A031001 Position of n-th 2 in A030998.

Original entry on oeis.org

2, 12, 21, 23, 25, 26, 27, 29, 31, 33, 40, 54, 68, 82, 99, 120, 134, 137, 140, 141, 143, 146, 149, 152, 162, 183, 204, 225, 238, 241, 244, 246, 247, 250, 253, 256, 259, 262, 265, 267, 268, 271, 274, 277, 280, 281, 283, 284, 286

Offset: 1

Clark Kimberling

nonn

A031002 Position of n-th 3 in A030998.

Original entry on oeis.org

3, 14, 28, 35, 37, 39, 41, 42, 43, 45, 47, 56, 70, 84, 102, 123, 144, 155, 158, 161, 164, 165, 167, 170, 173, 186, 207, 228, 249, 270, 291, 302, 305, 308, 311, 312, 314, 317, 320, 333, 354, 375, 385, 388, 391, 394, 396, 397, 400

Offset: 1

Clark Kimberling

nonn

A031003 Position of n-th 4 in A030998.

Original entry on oeis.org

4, 16, 30, 44, 49, 51, 53, 55, 57, 58, 59, 61, 72, 86, 105, 126, 147, 168, 176, 179, 182, 185, 188, 189, 191, 194, 210, 231, 252, 273, 294, 315, 323, 326, 329, 332, 335, 336, 338, 341, 357, 378, 399, 420, 441, 462, 470, 473, 476

Offset: 1

Clark Kimberling

nonn

Flatten[Position[Flatten[IntegerDigits[#,7]&/@Range[0,200]],4]]-1 (* Harvey P. Dale, May 06 2014 *)

A031004 Position of n-th 5 in A030998.

Original entry on oeis.org

5, 18, 32, 46, 60, 63, 65, 67, 69, 71, 73, 74, 75, 88, 108, 129, 150, 171, 192, 197, 200, 203, 206, 209, 212, 213, 215, 234, 255, 276, 297, 318, 339, 344, 347, 350, 353, 356, 359, 360, 362, 381, 402, 423, 444, 465, 486, 491, 494

Offset: 1

Clark Kimberling

nonn

A031005 Position of n-th 6 in A030998.

Original entry on oeis.org

6, 20, 34, 48, 62, 76, 77, 79, 81, 83, 85, 87, 89, 90, 111, 132, 153, 174, 195, 216, 218, 221, 224, 227, 230, 233, 236, 237, 258, 279, 300, 321, 342, 363, 365, 368, 371, 374, 377, 380, 383, 384, 405, 426, 447, 468, 489, 510, 512

Offset: 1

Clark Kimberling

nonn

Flatten[Position[Flatten[IntegerDigits[#,7]&/@Range[0,200]],6]]-1 (* Harvey P. Dale, Mar 25 2019 *)

A031006 a(n)=least k such that base 7 representation of n begins at s(k), where s=A030998.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 1, 13, 15, 17, 19, 21, 12, 25, 2, 29, 31, 33, 35, 14, 28, 41, 3, 45, 47, 49, 16, 30, 44, 57, 4, 61, 63, 18, 32, 46, 60, 73, 5, 77, 6, 20, 34, 48, 62, 76, 91, 7, 97, 100, 103, 106, 109, 96, 9, 10, 121, 124, 127, 130, 133, 11

Offset: 1

Clark Kimberling

nonn

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

A007376 The almost-natural numbers: write n in base 10 and juxtapose digits.

Original entry on oeis.org

0, 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, 7

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

Also called the Barbier infinite word.
This is an example of a non-morphic sequence.
a(n) = A162711(n,1); A136414(n) = 10*a(n) + a(n+1). - Reinhard Zumkeller, Jul 11 2009
a(A031287(n)) = 0, a(A031288(n)) = 1, a(A031289(n)) = 2, a(A031290(n)) = 3, a(A031291(n)) = 4, a(A031292(n)) = 5, a(A031293(n)) = 6, a(A031294(n)) = 7, a(A031295(n)) = 8, a(A031296(n)) = 9. - Reinhard Zumkeller, Jul 28 2011
May be regarded as an irregular table in which the n-th row lists the digits of n. - Jason Kimberley, Dec 07 2012
The digits of the integer n start at index A117804(n). The digit a(n) at index n belongs to the number A100470(n). - M. F. Hasler, Oct 23 2019
See also the Copeland-Erdős constant A033308, equivalent using primes instead of all numbers. - M. F. Hasler, Oct 24 2019
Decimal expansion of Sum_{k>=1} k/10^(A058183(k) + 1). - Stefano Spezia, Nov 30 2022

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, pp. 114, 336.
R. Honsberger, Mathematical Chestnuts from Around the World, MAA, 2001; see p. 163.
M. Kraitchik, Mathematical Recreations. Dover, NY, 2nd ed., 1953, p. 49.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 26.

Robert G. Wilson v, Table of n, a(n) for n = 0..100000 (a(0) = 0 added by _M. F. Hasler_, Oct 23 2019).
Putnam Competition No. 48, Problem A2, Math. Mag., 61 (1988), 131-134.
Robert G. Wilson v, Letter to N. J. A. Sloane, Oct. 1993

Considered as a sequence of digits, this is the same as the decimal expansion of the Champernowne constant, A033307. See that entry for a formula for a(n), further references, etc.
Cf. A054632 (partial sums), A023103.
For "decimations" see A127050 A127353 A127414 A127508 A127584 A127734 A127794 A127950 A128178 A128211 A128359 A128423 A128475 A128881.
Cf. A193428, A256100, A001477 (the nonnegative integers), A117804, A100470.
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), this sequence and A033307 (b=10). - Jason Kimberley, Dec 06 2012
Row lengths in A055642.
For primes here see A071620. See A007908 for a very similar sequence.
Cf. A031287, A031288, A031289, A031290, A031291, A031292, A031293, A031294, A031295, A031296, A033308, A058183, A136414, A162711.

a007376 n = a007376_list !! (n-1)
a007376_list = concatMap (map (read . return) . show) [0..] :: [Int]
-- Reinhard Zumkeller, Nov 11 2013, Dec 17 2011, Mar 28 2011

&cat[Reverse(IntegerToSequence(n)):n in[0..31]]; // Jason Kimberley, Dec 07 2012

c:=proc(x,y) local s: s:=proc(m) nops(convert(m,base,10)) end: if y=0 then 10*x else x*10^s(y)+y: fi end: b:=proc(n) local nn: nn:=convert(n,base,10):[seq(nn[nops(nn)+1-i],i=1..nops(nn))] end: A:=0: for n from 1 to 75 do A:=c(A,n) od: b(A); # c concatenates 2 numbers while b converts a number to the sequence of its digits - Emeric Deutsch, Jul 27 2006
#alternative
A007376 := proc(n) option remember ; local aprev, dOld,N ; if n <=9 then RETURN([n,n,1]) ; else aprev := A007376(n-1) ; dOld := op(3,aprev) ; N := op(2,aprev) ; if dOld < A055642(N) then RETURN([op(-dOld-1,convert(N,base,10)),N,dOld+1]) ; else RETURN([op(-1,convert(N+1,base,10)),N+1,1]) ; fi ; fi ; end: # R. J. Mathar, Jan 21 2008

Flatten[ IntegerDigits /@ Range@ 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] (* updated Jun 29 2014 *)
With[{nn=120},RealDigits[N[ChampernowneNumber[],nn],10,nn]][[1]] (* Harvey P. Dale, Mar 13 2018 *)

for(n=0,90,v=digits(n);for(i=1,#v,print1(v[i]", "))) \\ Charles R Greathouse IV, Nov 20 2012

apply( A007376(n)={for(k=1,n, k*10^k>n&& return(digits(n\k)[n%k+1]); n+=10^k)}, [0..200]) \\ M. F. Hasler, Nov 03 2019

A007376_list = [int(d) for n in range(10**2) for d in str(n)] # Chai Wah Wu, Feb 04 2015

Extended to a(0) = 0 by M. F. Hasler, Oct 23 2019

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A030999 Position of n-th 0 in A030998.

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A031000 Position of n-th 1 in A030998.

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A031001 Position of n-th 2 in A030998.

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A031002 Position of n-th 3 in A030998.

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A031003 Position of n-th 4 in A030998.

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A031004 Position of n-th 5 in A030998.

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A031005 Position of n-th 6 in A030998.

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A031006 a(n)=least k such that base 7 representation of n begins at s(k), where s=A030998.

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

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A007376 The almost-natural numbers: write n in base 10 and juxtapose digits.

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