A077772 -id:A077772 - cp's OEIS frontend

A244332 Position of the incrementally largest term in the continued fraction for the base 3 Champernowne constant (A077772).

1, 2, 4, 5, 7, 19, 39, 121, 321, 813, 2063, 4957, 11731, 26329, 59501, 132273, 294475, 644557, 1392933, 2982557

Offset: 1

John K. Sikora, Jun 26 2014

John K. Sikora, Table of n, a(n) for n = 1..20
J. K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT]
J. K. Sikora, Number of ternary digits of the first 2982556 terms of the continued fraction of the base 3 Champernowne Constant (7 MB zipped)

Cf. A077772 (Continued fraction expansion of the ternary Champernowne constant.)
Cf. A244757 (Incrementally largest terms in the continued fraction for the base 3 Champernowne constant.)
Cf. A244333 (Number of ternary digits in the high-water marks of the terms of the continued fraction of the base 3 Champernowne constant.)

A244757 Incrementally largest terms in the continued fraction for the base 3 Champernowne constant (A077772).

Original entry on oeis.org

0, 1, 2, 37, 162, 3068518062211324, 1079268324684171943515797470873767312825026176345571319042096689270

Offset: 1

John K. Sikora, Jul 05 2014

John K. Sikora, Table of n, a(n) for n = 1..7

Cf. A077772 (Continued fraction expansion of the ternary Champernowne constant.)

Cf. A244332 (Position of the incrementally largest term in the continued fraction for the ternary Champernowne constant.)

A244333 Number of ternary digits in the high-water marks of the terms of the continued fraction of the base 3 Champernowne constant (A077772).

Original entry on oeis.org

0, 1, 1, 4, 5, 33, 139, 515, 1809, 6181, 20759, 68871, 226333, 738089, 2391459, 7705867, 24711977, 78918957, 251105839

Offset: 1

John K. Sikora, Jun 27 2014

John K. Sikora, Table of n, a(n) for n = 1..19
J. K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT]
J. K. Sikora, Number of ternary digits of the first 2982556 terms of the continued fraction of the base 3 Champernowne Constant (7 MB zipped)

Cf. A077772 (Continued fraction expansion of the ternary Champernowne constant.)
Cf. A244757 (Incrementally largest terms in the continued fraction for the base 3 Champernowne constant.)
Cf. A244332 (Position of the incrementally largest term in the continued fraction for the base 3 Champernowne constant.)

puts (5..19).collect {|n| (1..(n-3)).inject(0) {|sum, m| sum+2*m*3**(m-1)}+3-n-2*((1..(n-4)).inject(0) {|sum1, m1| sum1+2*m1*3**(m1-1)}+3-(n-1))-3*(n-2)+4}

It appears that: Define NCD(N)=3-N+(sum{m=1..(N-3)} 2*m*3^(m-1)); then for n>=5, a(n) = NCD(n)-2*NCD(n-1)-3*(n-2)+4.

A244759 Number of ternary digits in the n-th term of the continued fraction of the base 3 Champernowne constant (A077772).

Original entry on oeis.org

0, 1, 1, 1, 4, 1, 5, 1, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 33, 1, 1, 1, 2, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 2, 2, 6, 1, 139, 1, 1, 1, 2, 2, 3, 2, 1, 3, 1, 1, 2, 4, 1, 2, 2, 1, 4, 1, 1, 2, 1, 1, 2, 2, 2, 3, 1, 1, 2, 2, 1, 1, 1

Offset: 1

John K. Sikora, Jul 06 2014

John K. Sikora, Table of n, a(n) for n = 1..4956
J. K. Sikora, Analysis of the High Water Mark Convergents of Champernowne's Constant in Various Bases, arXiv:1408.0261 [math.NT]
J. K. Sikora, Number of ternary digits of the first 2982556 terms of the continued fraction of the base 3 Champernowne Constant (7 MB zipped)

Cf. A077772 (Continued fraction expansion of the ternary Champernowne constant.)

Cf. A244332 (Position of the incrementally largest term in the continued fraction for the base 3 Champernowne constant.)

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

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

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

Joshua Searle, Dec 13 2024

OeisWiki, Champernowne constant

Cf. A030373 (base 4 expansion), A378328 (decimal expansion).

Other continued fractions: A066717, A077772, A378346, A378347, A378348, A378349, A378350, A030167.

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

Joshua Searle, Dec 13 2024

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

OeisWiki, Champernowne constant

Cf. A031219 (base 5 expansion), A378329 (decimal exapansion).

Other continued fractions: A066717, A077772, A378345, A378347, A378348, A378349, A378350, A030167.

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

Joshua Searle, Dec 13 2024

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

OeisWiki, Champernowne constant

Cf. A030548 (base 6 expansion), A378330 (decimal expansion).

Other continued fractions: A066717, A077772, A378345, A378346, A378348, A378349, A378350, A030167.

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

Joshua Searle, Dec 14 2024

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

OeisWiki, Champernowne constant

Cf. A030998 (base 7 expansion), A378331 (decimal expansion).

Other continued fractions: A066717, A077772, A378345, A378346, A378347, A378349, A378350, A030167.

ContinuedFraction[ChampernowneNumber[7], 100]

cp's OEIS Frontend

A244332 Position of the incrementally largest term in the continued fraction for the base 3 Champernowne constant (A077772).

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A244757 Incrementally largest terms in the continued fraction for the base 3 Champernowne constant (A077772).

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A244333 Number of ternary digits in the high-water marks of the terms of the continued fraction of the base 3 Champernowne constant (A077772).

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A244759 Number of ternary digits in the n-th term of the continued fraction of the base 3 Champernowne constant (A077772).

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

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A077771 Decimal value of the ternary Champernowne constant.

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PARI

A378345 Continued fraction expansion of the base 4 Champernowne constant.

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A378346 Continued fraction expansion of the base 5 Champernowne constant.

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A378347 Continued fraction expansion of the base 6 Champernowne constant.

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Mathematica

A378348 Continued fraction expansion of the base 7 Champernowne constant.

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