cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 19 results. Next

A244330 Position of the incrementally largest term in the continued fraction for the base-2 Champernowne constant (A066717).

Original entry on oeis.org

1, 2, 3, 15, 31, 65, 153, 343, 775, 1729, 3735, 8175, 17371, 36447, 76077, 157471, 324801, 669469, 1370903, 2807215, 5728545, 11679949, 23777059, 48348375, 98093505
Offset: 1

Views

Author

John K. Sikora, Jun 25 2014

Keywords

Links

Crossrefs

Cf. A066717 (The continued fraction for the "binary" Champernowne constant.)
Cf. A066718 (Incrementally largest terms in the continued fraction for the "binary" Champernowne constant.)
Cf. A244331 (Number of binary digits in the high-water marks of the terms of the continued fraction of the base 2 Champernowne constant.)

A244758 Number of binary digits in the n-th term of the continued fraction of the base-2 Champernowne constant (A066717).

Original entry on oeis.org

0, 1, 3, 2, 1, 3, 3, 2, 2, 1, 3, 3, 1, 2, 9, 1, 3, 1, 1, 2, 9, 3, 2, 3, 1, 2, 1, 2, 1, 1, 23, 1, 3, 3, 1, 2, 1, 3, 1, 5, 1, 3, 1, 3, 3, 1, 3, 1, 3, 1, 2, 1, 3, 2, 2, 6, 2, 4, 1, 4, 2, 3, 2, 5, 53, 1, 1, 7, 1, 3, 3, 3, 3
Offset: 1

Views

Author

John K. Sikora, Jul 05 2014

Keywords

Links

Crossrefs

Cf. A066717 (The continued fraction for the "binary" Champernowne constant.)
Cf. A244330 (Position of the incrementally largest term in the continued fraction for the base 2 Champernowne constant.)

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

Views

Author

Eric W. Weisstein, Nov 15 2002

Keywords

Links

Crossrefs

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.

Programs

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

A066716 Decimal expansion of the binary Champernowne constant 0.862240125868... whose binary expansion is the concatenation of 1, 2, 3, ... written in binary.

Original entry on oeis.org

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

Views

Author

Robert G. Wilson v, Jan 14 2002

Keywords

Comments

A theorem of Copeland & Erdős proves that this constant is 2-normal. - Charles R Greathouse IV, Feb 06 2015
This constant is transcendental. Note that this result is nontrivial: it is not a corollary of the result of Masaaki Amou saying that the base-b Champernowne constant has irrationality measure b, because the Thue-Siegel-Roth theorem only guarantees that a number with irrationality measure greater than 2 is transcendental. However, it is already stated in Masaaki Amou's paper that K. Mahler proved that the base-b Champernowne constant is transcendental for all b. - Jianing Song, Sep 27 2023

Examples

			0.8622401258680545715577902832493945785657647427682990945160712145573067405905...

Links

Crossrefs

Cf. A030302 (binary digits), A030190 (same with initial 0), A030303 (indices of 1's), A007088, A047778 (concatenate binary 1..n).
Cf. A066717 (continued fraction), A365238 (reciprocal).
Cf. A100125 (Sum n/2^(n^2)).
Cf. A033307.

Programs

  • Mathematica
    a = {}; Do[a = Append[a, IntegerDigits[n, 2]], {n, 1, 100} ]; RealDigits[ N[ FromDigits[ {Flatten[a], 0}, 2], 100]]
First[RealDigits[ChampernowneNumber[2], 10, 100]] (* Paolo Xausa, Jun 12 2024 *)
  • PARI
    my(s=0.); forstep(n=default(realprecision),1,-1,s=(s+n)>>#binary(n)); s \\ Charles R Greathouse IV, Feb 06 2015, corrected by M. F. Hasler, Mar 22 2017
  • PARI
    s=0;sum(n=1,31,n*.5^s+=logint(n,2)+1) \\ Accurate to 0.5^s. The sum up to n=31 is enough for standard precision of 38 digits. - M. F. Hasler, Mar 22 2017

Formula

The "binary" Champernowne constant is the number whose base-2 expansion is the concatenation of the binary representations of the integers, 0.(1)(10)(11)(100)(101)(110)(111)(1000)..., cf. A030302.

Extensions

Leading zero removed, offset adjusted, and keyword:cons added by R. J. Mathar, Mar 04 2010
Name edited by M. F. Hasler, Oct 26 2019

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

Views

Author

Eric W. Weisstein, Nov 15 2002

Keywords

Comments

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

Examples

			0.598958167538433992500172217929...

Links

Crossrefs

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.

Programs

  • Mathematica
    First[RealDigits[ChampernowneNumber[3], 10, 100]] (* Paolo Xausa, May 03 2024 *)
  • PARI
    A077771(b=3,t=1.,s=b)={sum(n=1, default(realprecision)*2.303\log(b)+1, nM. F. Hasler, Oct 25 2019

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

Views

Author

Joshua Searle, Dec 13 2024

Keywords

Links

Crossrefs

Cf. A030373 (base 4 expansion), A378328 (decimal expansion).
Other continued fractions: A066717, A077772, A378346, A378347, A378348, A378349, A378350, A030167.

Programs

  • Mathematica
    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

Views

Author

Joshua Searle, Dec 13 2024

Keywords

Comments

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

Links

Crossrefs

Cf. A031219 (base 5 expansion), A378329 (decimal exapansion).
Other continued fractions: A066717, A077772, A378345, A378347, A378348, A378349, A378350, A030167.

Programs

  • Mathematica
    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

Views

Author

Joshua Searle, Dec 13 2024

Keywords

Comments

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

Links

Crossrefs

Cf. A030548 (base 6 expansion), A378330 (decimal expansion).
Other continued fractions: A066717, A077772, A378345, A378346, A378348, A378349, A378350, A030167.

Programs

  • Mathematica
    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

Views

Author

Joshua Searle, Dec 14 2024

Keywords

Comments

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

Links

Crossrefs

Cf. A030998 (base 7 expansion), A378331 (decimal expansion).
Other continued fractions: A066717, A077772, A378345, A378346, A378347, A378349, A378350, A030167.

Programs

  • Mathematica
    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

Views

Author

Joshua Searle, Dec 14 2024

Keywords

Comments

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

Links

Crossrefs

Cf. A054634 (base 8 expansion), A378332 (decimal expansion).
Other continued fractions: A066717, A077772, A378345, A378346, A378347, A378348, A378350, A030167.

Programs

  • Mathematica
    ContinuedFraction[ChampernowneNumber[8], 100]
Showing 1-10 of 19 results. Next

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