A066716 -id:A066716 - cp's OEIS frontend

A365238 Decimal expansion of 1/A066716 (Binary Champernowne constant).

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

Offset: 1

Kelvin Voskuijl, Aug 27 2023

			1.15976973235066807158695812033014828339074163177157159329786...

Cf. A066716.

RealDigits[1/ChampernowneNumber[2] , 10, 120][[1]]

A033308 Decimal expansion of Copeland-Erdős constant: concatenate primes.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein

The number .23571113171923.... was proved normal in base 10 by Copeland and Erdős but is not known to be normal in other bases. - Jeffrey Shallit, Mar 14 2008
Could be read (with indices 1, 2, ...) as irregular table whose n-th row lists the A097944(n) digits of the n-th prime A000040(n). - M. F. Hasler, Oct 25 2019
Named after the American mathematician Arthur Herbert Copeland (1898-1970) and the Hungarian mathematician Paul Erdős (1913-1996). - Amiram Eldar, May 29 2021
This constant is irrational but it is not (yet) known to be transcendental. - Charles R Greathouse IV, Feb 03 2025

			0.235711131719232931374143475359616771737983899710110310710911312...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.9, p. 442.
Glyn Harman, One hundred years of normal numbers, in M. A. Bennett et al., eds., Number Theory for the Millennium, II (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 149-166.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

T. D. Noe, Table of n, a(n) for n = 0..2000
John M. Campbell, The prime-counting Copeland-Erdős constant, arXiv:2309.13520 [math.NT], 2023.
A. H. Copeland and P. Erdős, Note on Normal Numbers, Bull. Amer. Math. Soc., Vol. 52, No. 10 (1946), pp. 857-860.
Mikołaj Morzy, Tomasz Kajdanowicz, and Przemysław Kazienko, On Measuring the Complexity of Networks: Kolmogorov Complexity versus Entropy, Complexity, Volume 2017 (2017), Article ID 3250301, p. 5.
Simon Plouffe, Copeland-Erdos constant, the primes concatenated.
Simon Plouffe, Copeland-Erdos constant, the primes concatenated.
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant.

Cf. A030168 (continued fraction), A072754 (numerators of convergents), A072755 (denominators of convergents).
Cf. A000040 (primes), A097944 (row lengths if this is read as table), A228355 (digits of the primes listed in reversed order).
Cf. A033307 (Champernowne constant: analog for positive integers instead of primes), A007376 (digits of the integers, considered as infinite word or table), A066716 (decimals of the binary Champernowne constant).
Cf. A165450, A019518, A074721, A073034, A191232, A129808.
Cf. A066747 and A191232: binary Copeland-Erdős constant: decimals and binary digits.
See also A338072.

a033308 n = a033308_list !! (n-1)
a033308_list = concatMap (map (read . return) . show) a000040_list :: [Int]
-- Reinhard Zumkeller, Mar 03 2014

N[Sum[Prime[n]*10^-(n + Sum[Floor[Log[10, Prime[k]]], {k, 1, n}]), {n, 1, 40}], 100] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006 *)
N[Sum[Prime@n*10^-(n + Sum[Floor[Log[10, Prime@k]], {k, n}]), {n, 45}], 106] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006 *)
IntegerDigits //@ Prime@Range@45 // Flatten (* Robert G. Wilson v Oct 03 2006 *)

default(realprecision, 2080); x=0.0; m=-1; forprime (p=2, 4000, n=1+floor(log(p)/log(10)); x=p+x*10^n; m+=n; ); x=x/10^m; for (n=0, 2000, d=floor(x); x=(x-d)*10; write("b033308.txt", n, " ", d)); \\ Harry J. Smith, Apr 30 2009

concat( apply( {row(n)=digits(prime(n))},  [1..99] )) \\ Yields this sequence; row(n) then yields the digits of prime(n) = n-th row of the table, cf. comments. - M. F. Hasler, Oct 25 2019

Equals Sum_{n>=1} prime(n)*10^(-A068670(n)). - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006

Equals Sum_{i>=1} (p_i * 10^(-(Sum_{j=1..i} 1 + floor(log_10(p_j))) )) or Sum_{i>=1} (p_i * 10^(-( i + Sum_{j=1..i} floor(log_10(p_j))) )) or Sum_{i>=1} (p_i * 10^(-( Sum_{j=1..i} ceiling(log_10(1 + p_j))) )). - Daniel Forgues, Mar 26-28 2014

A047778 Concatenation of the first n numbers in binary (converted to base 10).

Original entry on oeis.org

1, 6, 27, 220, 1765, 14126, 113015, 1808248, 28931977, 462911642, 7406586283, 118505380540, 1896086088653, 30337377418462, 485398038695407, 15532737238253040, 497047591624097297, 15905522931971113522, 508976733823075632723, 16287255482338420247156

Offset: 1

Aaron Gulliver (gulliver(AT)elec.canterbury.ac.nz)

The smallest prime in this sequence is 485398038695407. What is the full subsequence of primes? - N. J. A. Sloane, Oct 03 2015

There is only the one prime in the first 22400 terms, making a second prime > 10^91000. - Hans Havermann, Oct 07 2015

			a(4) = 1 10 11 100 [base 2] = 220 [base 10].

Joe B. Stephen, Table of n, a(n) for n = 1..400 (terms 1..250 from Reinhard Zumkeller)

Cf. A001855 (bit counts, offset by 1), A061168, A066716.

Concatenation of first n numbers in other bases: 2: this sequence, 3: A048435, 4: A048436, 5: A048437, 6: A048438, 7: A048439, 8: A048440, 9: A048441, 10: A007908, 11: A048442, 12: A048443, 13: A048444, 14: A048445, 15: A048446, 16: A048447.

a047778 = (foldl (\v d -> 2*v + d) 0) . concatMap (reverse . unfoldr
   (\x -> if x == 0 then Nothing else Just $ swap $ divMod x 2)) .
   enumFromTo 1
-- Reinhard Zumkeller, Feb 19 2012

conc:= (x,y) -> x*2^(1+ilog2(y))+y:
a[1]:= 1:
for n from 2 to 30 do a[n]:= conc(a[n-1],n) od:
seq(a[n],n=1..30); # Robert Israel, Oct 07 2015

If[STARTPOINT==1,n={},n=Flatten[IntegerDigits[Range[STARTPOINT-1],2]]]; Table[AppendTo[n,IntegerDigits[w,2]];n=Flatten[n];FromDigits[n,2],{w,STARTPOINT,ENDPOINT}] (* Dylan Hamilton, Aug 04 2010 *)
f[n_] := FromDigits[ Flatten@ IntegerDigits[ Range@n, 2], 2]; Array[f, 18] (* Robert G. Wilson v, Nov 07 2010 *)
Module[{n = 1}, NestList[#*2^BitLength[++n] + n &, 1, 25]] (* Paolo Xausa, Sep 30 2024 *)

cb(a,b)=a<<#binary(b) + b
a(n)=fold(cb, [1..n]) \\ Charles R Greathouse IV, Jun 21 2017

A047778_vec(N=20,s)=vector(N,k,s=s<M. F. Hasler, Oct 25 2019

def a(n): return int("".join([(bin(i))[2:] for i in range(1, n+1)]), 2)
print([a(n) for n in range(1, 19)]) # Michael S. Branicky, Jan 06 2021

from functools import reduce
def A047778(n): return reduce(lambda i,j:(i<Chai Wah Wu, Feb 26 2023

a(n) = a(n-1)*2^(1+floor(log_2(n))) + n. - Henry Bottomley, Jan 12 2001

a(n) = 4C / 2^frac(log_2(n)) * n^{n+1} / r(frac(log_2(n)))^n + O(1), where r(x) = 2^{x - 1 + 2^{1-x}}; frac is the fractional part function frac(x) = x - floor(x); and C is the binary Champernowne constant (A066716). (In fact, a(n) is the floor of this expression; the error term is between 1/2 and 1.) r(x) takes on values between e*log(2) and 2 for x in the range 0 to 1. It follows using Stirling's approximation that the radius of convergence for the e.g.f. is log 2. - Franklin T. Adams-Watters, Sep 07 2006

More terms from Patrick De Geest, May 15 1999

Name edited by Joe B. Stephen, Jul 22 2023

A066717 The continued fraction for the "binary" Champernowne constant.

Original entry on oeis.org

0, 1, 6, 3, 1, 6, 5, 3, 3, 1, 6, 4, 1, 3, 298, 1, 6, 1, 1, 3, 285, 7, 2, 4, 1, 2, 1, 2, 1, 1, 4534532, 1, 4, 5, 1, 2, 1, 7, 1, 16, 1, 4, 1, 5, 5, 1, 5, 1, 4, 1, 2, 1, 5, 3, 2, 38, 2, 12, 1, 15, 2, 6, 3, 30, 4682854730443938, 1, 1, 68, 1, 6, 5, 4, 4, 1, 2, 1, 1, 1, 1, 2, 22, 1, 2, 7, 1, 2

Offset: 0

Robert G. Wilson v, Jan 14 2002

Robert G. Wilson v, Table of n, a(n) for n = 0..1000
J. K. Sikora, The first 98093504 CFE coefficients of the binary Champernowne Constant (231 MB zipped)
Eric E. Weisstein, Binary Champernowne Constant

Cf. A030190 & A066716 (binary & decimal digits of the binary Champernowne constant), A033307 (decimal Champernowne constant).

Cf. A054635, A077771, A077772: base 3, decimals and continued fraction of ternary Champernowne constant.

a = {}; Do[a = Append[a, IntegerDigits[n, 2]], {n, 1, 10^3} ]; ContinuedFraction[ N[ FromDigits[ {Flatten[a], 0}, 2], 500]]
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]]]; Take[ ContinuedFraction[ FromDigits[ {Array[almostNatural[#, 2] &, 20000], 0}, 2]], 100] (* Robert G. Wilson v, Jul 21 2014 *)

A066717(b=2,t=1.,s=b)={contfrac(sum(n=1,default(realprecision)*2.303\log(b)+1, nM. F. Hasler, Oct 25 2019

A077772 Continued fraction expansion of the ternary Champernowne constant.

Original entry on oeis.org

0, 1, 1, 2, 37, 1, 162, 1, 1, 1, 3, 1, 7, 1, 9, 2, 3, 1, 3068518062211324, 2, 1, 2, 6, 13, 1, 2, 1, 3, 1, 10, 1, 21, 1, 1, 4, 3, 577, 1, 1079268324684171943515797470873767312825026176345571319042096689270, 1, 1, 1, 3, 4, 21, 3, 1, 9, 1

Offset: 0

Eric W. Weisstein, Nov 15 2002

John K. Sikora, Table of n, a(n) for n = 0..2061 (terms n = 0..1155 from Robert G. Wilson v)
J. K. Sikora, The first 2982556 terms of the CFE of the ternary Champernowne Constant (141 MB zipped)
Eric Weisstein's World of Mathematics, Ternary Champernowne Constant

Cf. A054635 (ternary digits), A077771 (decimals).

Cf. A030190, A066716, A066717: binary digits, decimals and continued fraction of the binary Champernowne constant; A033307: decimal Champernowne constant.

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]]]; Take[ ContinuedFraction[ FromDigits[ {Array[almostNatural[#, 3] &, 20000], 0}, 3]], 100] (* Robert G. Wilson v, Jul 21 2014 *)

\p 10000
t=0;r=0.;T=1; for(n=1,1e6,d=#digits(n,3);t+=d;T*=3^d;r+=n/T;if(t>20959, return)); v=contfrac(r); v[1..30] \\ Charles R Greathouse IV, Sep 23 2014

A077772(b=3,t=1.,s=b)={contfrac(sum(n=1,default(realprecision)*2.303/log(b)+1, nM. F. Hasler, Oct 25 2019

A058935 Concatenation of first n binary numbers.

Original entry on oeis.org

0, 1, 110, 11011, 11011100, 11011100101, 11011100101110, 11011100101110111, 110111001011101111000, 1101110010111011110001001, 11011100101110111100010011010, 110111001011101111000100110101011, 1101110010111011110001001101010111100

Offset: 0

Henry Bottomley, Jan 12 2001

If the terms are read as decimal numbers, which of them are primes? For example, a(5) = 11011100101 = 1193*9229757 is not a prime. - N. J. A. Sloane, Feb 17 2023

Answer: a(231) is the first prime term when read as a decimal number; a(15) is the first when read as a binary number. - Michael S. Branicky, Feb 17 2023

Michael S. Branicky, Table of n, a(n) for n = 0..155
Eric Weisstein's World of Mathematics, Binary Champernowne Constant
Eric Weisstein's World of Mathematics, Smarandache Number
Index entries for sequences related to Most Wanted Primes video

Cf. A047778 for this converted to decimal, A001855 (offset) for number of digits.
Cf. A066716: binary Champernowne constant, A030302: binary digits, A030190: same with initial 0, A030303: indices of 1's, A007088.
Other bases: A117640 (4), A007908 (10).

FromDigits /@ Flatten /@ Rest[FoldList[Append, {}, IntegerDigits[Range[10], 2]]] (* Eric W. Weisstein, Nov 04 2015 *)

from itertools import count, islice
def agen(s=""): yield from (int(s:=s+bin(n)[2:]) for n in count(0))
print(list(islice(agen(), 13))) # Michael S. Branicky, Feb 17 2023

from functools import reduce
def A058935(n): return int(bin(reduce(lambda i,j:(i<Chai Wah Wu, Feb 26 2023

a(n) = a(n-1)*10^A029837(n) + A007088(n).

A077771 Decimal value of the ternary Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Nov 15 2002

The first 99 digits form a prime. - Jonathan Vos Post, Nov 11 2004

This constant is 3-normal. - Charles R Greathouse IV, Feb 06 2015

			0.598958167538433992500172217929...

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Ternary Champernowne Constant

Cf. A054635 (base 3 digits), A077772 (continued fraction).

Cf. A030190, A066716, A066717: binary digits, decimals and continued fraction of the binary Champernowne constant; A033307: decimal Champernowne constant.

First[RealDigits[ChampernowneNumber[3], 10, 100]] (* Paolo Xausa, May 03 2024 *)

A077771(b=3,t=1.,s=b)={sum(n=1, default(realprecision)*2.303\log(b)+1, nM. F. Hasler, Oct 25 2019

A378328 Decimal expansion of the base 4 Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Joshua Searle, Nov 23 2024

This constant is formed by the concatenation of the natural numbers in base 4 and then converted into base 10.

This constant is 4-normal.

			0.426111111111111065764556571420161985095546238967230410682791635172587553...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[4], 10, 100]]

A378329 Decimal expansion of the base 5 Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Joshua Searle, Nov 23 2024

This constant is formed by the concatenation of the natural numbers in base 5 and then converted into base 10.

This constant is 5-normal.

			0.310736111111111111111111111110963033311604944849115504682622268470343392...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[5], 10, 100]]

A378330 Decimal expansion of the base 6 Champernowne constant.

Original entry on oeis.org

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

Offset: 0

Joshua Searle, Nov 23 2024

This constant is formed by the concatenation of the natural numbers in base 6 and then converted into base 10.

This constant is 6-normal.

			0.239862685815066767447719828672209624590576971529350213760693195631576583...

OeisWiki, Champernowne constant

Cf. A030302, A003137, A030373, A031219, A030548, A030998, A054634, A031076, A033307 (base n expansions of base n Champernowne constants, without leading zero, for 2 <= n <= 10).
Cf. A066716, A077771, A378328, A378329, A378330, A378331, A378332, A378333, A033307 (decimal expansions of base n Champernowne constants for 2 <= n <= 10).
Cf. A066717, A077772, A378345, A378346, A378347, A378348, A378349, A378350, A030167 (continued fraction expansions of base n Champernowne constants for 2 <= n <= 10).

First[RealDigits[ChampernowneNumber[6], 10, 100]]

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A365238 Decimal expansion of 1/A066716 (Binary Champernowne constant).

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A033308 Decimal expansion of Copeland-Erdős constant: concatenate primes.

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A047778 Concatenation of the first n numbers in binary (converted to base 10).

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A066717 The continued fraction for the "binary" Champernowne constant.

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A077772 Continued fraction expansion of the ternary Champernowne constant.

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A058935 Concatenation of first n binary numbers.

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A077771 Decimal value of the ternary Champernowne constant.

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