A001191 -id:A001191 - 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

A134724 Successive decimal digits of the cubes A000578.

Original entry on oeis.org

0, 1, 8, 2, 7, 6, 4, 1, 2, 5, 2, 1, 6, 3, 4, 3, 5, 1, 2, 7, 2, 9, 1, 0, 0, 0, 1, 3, 3, 1, 1, 7, 2, 8, 2, 1, 9, 7, 2, 7, 4, 4, 3, 3, 7, 5, 4, 0, 9, 6, 4, 9, 1, 3, 5, 8, 3, 2, 6, 8, 5, 9, 8, 0, 0, 0, 9, 2, 6, 1, 1, 0, 6, 4, 8, 1, 2, 1, 6, 7, 1, 3, 8, 2, 4, 1, 5, 6, 2, 5, 1, 7, 5, 7, 6, 1, 9, 6, 8, 3

Offset: 0

Omar E. Pol, Nov 12 2007

Davenport & Erdős show that 0.0182764125..., this sequence interpreted as a constant, is 10-normal. (Indeed, it shows this property for any nonconstant polynomial mapping positive integers to positive integers has this property, regardless of the base chosen.) - Charles R Greathouse IV, Oct 02 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..8000
H. Davenport and P. Erdős, Note on normal decimals, Canadian Journal of Mathematics 4 (1952), pp. 58-63.

Cf. A007376, A001191.

Flatten[IntegerDigits/@(Range[0,30]^3)] (* Harvey P. Dale, Sep 12 2012 *)

for(n=0,10,my(d=digits(n^3));for(i=1,#d,print1(d[i]", "))) \\ Charles R Greathouse IV, Oct 02 2013

A340207 Constant whose decimal expansion is the concatenation of the largest n-digit square A061433(n), for n = 1, 2, 3, ...

Original entry on oeis.org

9, 8, 1, 9, 6, 1, 9, 8, 0, 1, 9, 9, 8, 5, 6, 9, 9, 8, 0, 0, 1, 9, 9, 9, 8, 2, 4, 4, 9, 9, 9, 8, 0, 0, 0, 1, 9, 9, 9, 9, 5, 0, 8, 8, 4, 9, 9, 9, 9, 8, 0, 0, 0, 0, 1, 9, 9, 9, 9, 9, 5, 1, 5, 5, 2, 9, 9, 9, 9, 9, 9, 8, 0, 0, 0, 0, 0, 1, 9, 9, 9, 9, 9, 9, 5, 8

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A339978 converge to this sequence of digits, and to this constant, up to powers of 10.

			The largest square with 1, 2, 3, 4, ... digits is, respectively, 9 = 3^2, 81 = 9^2, 961 = 31^2, 9801 = 99^2, ....
Here we list the sequence of digits of these numbers: 9; 8, 1; 9, 6, 1; 9, 8, 0, 1; 9, 9, 8, 5, 6; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.98196198...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A061433 (largest n-digit square), A339978 (has this as "limit"), A340208 (same with "smallest n-digit cube", limit of A215692), A340209 (same for cubes, limit of A340115), A340220 (same for primes), A340219 (similar, with smallest primes, limit of A215641), A340222 (same for semiprimes), A340221 (same for smallest semiprimes, limit of A215647).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

lnds[k_]:=Module[{c=Sqrt[10^k]},If[IntegerQ[c],(c-1)^2,Floor[c]^2]]; Flatten[IntegerDigits/@(lnds/@Range[15])] (* Harvey P. Dale, Dec 16 2021 *)

concat([digits(sqrtint(10^k-1)^2)|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*sqrtint(10^k-1)^2) \\ as constant

c = 0.9819619801998569980019998244999800019999508849999800001999995155...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*floor(10^(k/2)-1)^2
a(-n(n+1)/2) = 9 for all n >= 2.

A340208 Constant whose decimal expansion is the concatenation of the smallest n-digit cube A061434(n), for n = 1, 2, 3, ...

Original entry on oeis.org

1, 2, 7, 1, 2, 5, 1, 0, 0, 0, 1, 0, 6, 4, 8, 1, 0, 3, 8, 2, 3, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 7, 7, 6, 9, 6, 1, 0, 0, 5, 4, 4, 6, 2, 5, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 7, 8, 7, 3, 8, 7, 5, 1, 0, 0, 0, 2, 6, 5, 7, 7, 2, 8, 8, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

M. F. Hasler, Dec 31 2020

Every third smallest n-digit cube (i.e., for n = 3k + 1, k >= 0) is 10^k, which explains the chunks of (1,0,...,0), cf. formula.

The terms of sequence A215692 converge to this sequence of digits, and to this constant, up to powers of 10.

			The smallest cube with 1, 2, 3, 4, ... digits is, respectively, 1, 27 = 3^3, 125 = 5^3, 1000 = 10^3, .... Here we list the sequence of digits of these numbers: 1; 2, 7; 1, 2, 5; 1, 0, 0, 0; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.1271251000106481...

Cf. A061434 (smallest n-digit cube), A215692 (has this as "limit"), A340209 (same with largest n-digit cubes, limit of A340115), A340206 (same for squares, limit of A215689), A340219 (same for primes, limit of A215641), A340221 (same for semiprimes, limit of A215647).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

concat([digits(ceil(10^((k-1)/3))^3)|k<-[1..14]]) \\ as seq. of digits
c(N=12)=sum(k=1,N,.1^(k*(k+1)/2)*ceil(10^((k-1)/2))^2) \\ as constant

c = 0.12712510001064810382310000001007769610054462510000000001000787387510002657...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*ceiling(10^((k-1)/3))^2
a(-n(n+1)/2) = 1 for all n >= 2;
a(k) = 0 for -3n(3n+1)/2 > k > -(3n+1)(3n+2)/2, n >= 0.

A340219 Constant whose decimal expansion is the concatenation of the smallest n-digit prime A003617(n), for n = 1, 2, 3, ...

Original entry on oeis.org

2, 1, 1, 1, 0, 1, 1, 0, 0, 9, 1, 0, 0, 0, 7, 1, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 1, 9, 1, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 9, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 7, 1, 0, 0, 0

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A215641 converge to this sequence of digits, and to this constant, up to powers of 10.

			The smallest prime with 1, 2, 3, 4, ... digits is, respectively, 2, 11, 101, 1009, .... Here we list the sequence of digits of these numbers: 2: 1, 1; 1, 0, 1; 1, 0, 0, 9; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.2111011009...

Cf. A003617 (smallest n-digit prime), A215641 (has this as "limit"), A340206 (same for squares, limit of A215689), A340207 (similar, with largest n-digit squares, limit of A339978), A340208 (same for cubes, limit of A215692), A340209 (same with largest n-digit cube, limit of A340115), A340221 (same for semiprimes, limit of A215647).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

Flatten[Table[IntegerDigits[NextPrime[10^n]],{n,0,20}]] (* Harvey P. Dale, Mar 29 2024 *)

concat([digits(nextprime(10^k))|k<-[0..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*nextprime(10^(k-1))) \\ as constant

c = 0.21110110091000710000310000031000001910000000710000000071000000001...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*nextprime(10^(k-1))
a(-n(n+1)/2) = 1 for all n >= 2, followed by increasingly more zeros.

A340221 Constant whose decimal expansion is the concatenation of the smallest n-digit semiprime A098449(n), for n = 1, 2, 3, ...

Original entry on oeis.org

4, 1, 0, 1, 0, 6, 1, 0, 0, 3, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A215647 converge to this sequence of digits, and to this constant, up to powers of 10.

			The smallest prime with 1, 2, 3, 4, ... digits is, respectively, 2, 11, 101, 1009, .... Here we list the sequence of digits of these numbers: 2: 1, 1; 1, 0, 1; 1, 0, 0, 9; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.2111011009...

Cf. A098449 (smallest n-digit semiprime), A215647 (has this as "limit"), A340206 (same for squares, limit of A215689), A340207 (similar, with largest n-digit squares, limit of A339978), A340208 (same for cubes, limit of A215692), A340209 (same with largest n-digit cube, limit of A340115), A340219 (same for primes, limit of A215641).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

concat([digits(A098449(k))|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*A098449(k)) \\ as constant

c = 0.410106100310001100001100000110000001100000001100000000610000000003...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*A098449(k)
a(-n(n+1)/2) = 1 for all n >= 2, followed by increasingly more zeros.

A340209 Constant whose decimal expansion is the concatenation of the largest n-digit cube A061435(n), for n = 1, 2, 3, ...

Original entry on oeis.org

8, 6, 4, 7, 2, 9, 9, 2, 6, 1, 9, 7, 3, 3, 6, 9, 7, 0, 2, 9, 9, 9, 9, 3, 8, 3, 7, 5, 9, 9, 8, 9, 7, 3, 4, 4, 9, 9, 7, 0, 0, 2, 9, 9, 9, 9, 9, 9, 3, 9, 4, 8, 2, 6, 4, 9, 9, 9, 6, 1, 9, 4, 6, 7, 2, 1, 9, 9, 9, 7, 0, 0, 0, 2, 9, 9, 9, 9, 9, 9, 9, 9, 5

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A340115 converge to this sequence of digits, and to this constant, up to powers of 10.

			The largest cube with 1, 2, 3, 4, ... digits is, respectively, 8 = 2^3, 64 = 4^3, 729 = 9^3, 9261 = 21^3, ..., cf. A061435.
Here we list the sequence of digits of these numbers: 8; 6, 4; 7, 2, 9; 9, 2, 6, 1; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.864729926...

Cf. A061435 (largest n-digit cube), A340115 (has this as "limit"), A340208 (similar, with smallest n-digit cubes, limit of A215692), A340207 (same for squares, limit of A339978), A340220 (same for primes), A340222 (same for semiprimes), A340219 (similar, with smallest primes, limit of A215641), A340221 (similar, with smallest semiprimes, limit of A215647).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

concat([digits(sqrtnint(10^k-1,3)^3)|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*sqrtnint(10^k-1,3)^3) \\ as constant

c = 0.86472992619733697029999383759989734499700299999939482649996194672...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*floor(10^(k/3)-1)^3
a(-n(n+1)/2) = 9 for all n >= 3;

cp's OEIS Frontend

A067113 Let N = 0.149162536496481100121... = A001191, the concatenation of the squares. Then a(n) = sum of first n digits of N.

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A175959 Positions of 1's in A001191 (Digits of squares).

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A175960 First occurrences of decimal digits 0..9 in A001191 (Digits of squares).

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

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A134724 Successive decimal digits of the cubes A000578.

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A340207 Constant whose decimal expansion is the concatenation of the largest n-digit square A061433(n), for n = 1, 2, 3, ...

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A340208 Constant whose decimal expansion is the concatenation of the smallest n-digit cube A061434(n), for n = 1, 2, 3, ...

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A340219 Constant whose decimal expansion is the concatenation of the smallest n-digit prime A003617(n), for n = 1, 2, 3, ...

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