A054635 -id:A054635 - cp's OEIS frontend

A003137 Write n in base 3 and juxtapose.

1, 2, 1, 0, 1, 1, 1, 2, 2, 0, 2, 1, 2, 2, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 2, 2, 2, 0, 0, 2, 0, 1, 2, 0, 2, 2, 1, 0, 2, 1, 1, 2, 1, 2, 2, 2, 0, 2, 2, 1, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 0, 1, 0, 2, 1, 1, 0, 2, 2, 1

Offset: 1

N. J. A. Sloane

An irregular table in which the n-th row lists the base-3 digits of n, see A007089. - Jason Kimberley, Dec 07 2012

The base-3 Champernowne constant (A077771): it is normal in base 3. - Jason Kimberley, Dec 07 2012

			1,
2,
1,0,
1,1,
1,2,
2,0,
2,1,
2,2,
1,0,0,
1,0,1,.... _R. J. Mathar_, Aug 16 2021

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened
Eric Weisstein's World of Mathematics, Ternary.
Wikipedia, Ternary numeral system

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), this sequence 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 A033307 (b=10). - Jason Kimberley, Dec 06 2012

Cf. A081604 (row lengths), A053735 (row sums), A030341 (rows reversed), A077771, A007089.

a003137 n k = a003137_tabf !! (n-1) !! k
a003137_row n = a003137_tabf !! (n-1)
a003137_tabf = map reverse $ tail a030341_tabf
a003137_list = concat a003137_tabf
-- Reinhard Zumkeller, Feb 21 2013

&cat[Reverse(IntegerToSequence(n,3)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten@ IntegerDigits[ Range@ 40, 3] (* 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[ a[#, 3] &, 105] (* Robert G. Wilson v, Jul 01 2014 *)

from itertools import count, islice
from sympy.ntheory.factor_ import digits
def A003137_gen(): return (d for m in count(1) for d in digits(m,3)[1:])
A003137_list = list(islice(A003137_gen(),30)) # Chai Wah Wu, Jan 07 2022

More terms from Larry Reeves (larryr(AT)acm.org), Sep 25 2000

A030373 Write n in base 4 and juxtapose.

Original entry on oeis.org

1, 2, 3, 1, 0, 1, 1, 1, 2, 1, 3, 2, 0, 2, 1, 2, 2, 2, 3, 3, 0, 3, 1, 3, 2, 3, 3, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 2, 0, 1, 2, 1, 1, 2, 2, 1, 2, 3, 1, 3, 0, 1, 3, 1, 1, 3, 2, 1, 3, 3, 2, 0, 0, 2, 0, 1, 2, 0, 2, 2, 0, 3, 2, 1, 0

Offset: 1

Clark Kimberling

An irregular table in which the n-th row lists the base-4 digits of n. - Jason Kimberley, Nov 26 2012

The base-4 Champernowne constant: it is normal in base 4. - Jason Kimberley, Nov 26 2012

			1;
2;
3;
1,0;
1,1;
1,2;
1,3;
2,0;
2,1;
2,2;
...
3,3;
1,0,0;
1,0,1;
1,0,2;
1,0,3;
1,1,0; ....

R. J. Mathar, Table of n, a(n) for n = 1..4664
F. J. Aragon Artacho, D. H. Bailey, J. M. Borwein, and P. B. Borwein, Walking on real numbers, preprint September 2012.

Cf. A007376, A003137, A007090.

Tables in which the n-th row lists the base b digits of n: A030190 and A030302 (b=2), A003137 and A054635 (b=3), this sequence (b=4), A031219 (b=5), A030548 (b=6), A030998 (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

&cat[Reverse(IntegerToSequence(n,4)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten[IntegerDigits[Range[40],4]] (* Harvey P. Dale, Aug 23 2011 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A030373_gen(): return chain.from_iterable(digits(m, 4)[1:] for m in count(1))
A030373_list = list(islice(A030373_gen(), 30)) # Chai Wah Wu, Jan 07 2022

A031076 Write n in base 9 and juxtapose.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

An irregular table in which the n-th row lists the base-9 digits of n. - Jason Kimberley, Dec 07 2012

The base-9 Champernowne constant: it is normal in base 9. - Jason Kimberley, Dec 07 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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), this sequence (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

Cf. A031087, A007095.

a031076 n = a031076_list !! (n-1)
a031076_list = concat $ map reverse $ tail a031087_tabf
-- Reinhard Zumkeller, Jul 07 2015

&cat[Reverse(IntegerToSequence(n,9)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten@ IntegerDigits[ Range@ 55, 9] (* 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[#, 9] &, 105] (* Robert G. Wilson v, Jul 01 2014 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A031076_gen(): return chain.from_iterable(digits(m,9)[1:] for m in count(1))
A031076_list = list(islice(A031076_gen(),30)) # Chai Wah Wu, Jan 07 2022

A030998 Write n in base 7 and juxtapose.

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

An irregular table in which the n-th row lists the base-7 digits of n. - Jason Kimberley, Dec 07 2012

The base-7 Champernowne constant: it is normal in base 7. - Jason Kimberley, Dec 07 2012

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), this sequence (b=7), A031035 and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

Cf. A007093.

[0]cat &cat[Reverse(IntegerToSequence(n,7)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten[IntegerDigits[#,7]&/@Range[0,60]]  (* Harvey P. Dale, Mar 04 2011 *)
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[#, 7] &, 105, 0] (* Robert G. Wilson v, Jun 29 2014 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A030998_gen(): return chain.from_iterable(digits(m, 7)[1:] for m in count(0))
A030998_list = list(islice(A030998_gen(), 30)) # Chai Wah Wu, Jan 07 2022

A031035 Write n in base 8 and juxtapose.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

Apart from the initial term, identical to A054634.
Should not be merged with A054634 because there are many sequences which depend on this sequence starting with a 1. - N. J. A. Sloane, Jan 30 2010
An irregular table in which the n-th row lists the base-8 digits of n. - Jason Kimberley, Dec 07 2012
The base-8 Champernowne constant: it is normal in base 8. - Jason Kimberley, Dec 07 2012

Robert Israel, Table of n, a(n) for n = 1..10000

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), this sequence and A054634 (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

Cf. A007094.

&cat[Reverse(IntegerToSequence(n,8)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

seq(op(ListTools:-Reverse(convert(n,base,8))),n=1..100); # Robert Israel, Nov 12 2024

Flatten[ IntegerDigits[ Range[40], 8]] (* 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[#, 8] &, 105] (* Robert G. Wilson v, Jun 29 2014 *)

from itertools import count, islice
from sympy.ntheory.factor_ import digits
def A031035_gen(): return (d for m in count(1) for d in digits(m,8)[1:])
A031035_list = list(islice(A031035_gen(),30)) # Chai Wah Wu, Jan 07 2022

A054634 Champernowne sequence: write n in base 8 and juxtapose.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Apr 16 2000

Apart from the initial term, identical to A031035.
Should not be merged with A031035 because there are many sequences which depend on the latter starting with a 1. - N. J. A. Sloane, Jan 30 2010
An irregular table in which the n-th row lists the base-8 digits of n. - Jason Kimberley, Dec 07 2012
The base-8 Champernowne constant: it is normal in base 8. - Jason Kimberley, Dec 07 2012

Cf. A007376, A030190, A007094.

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 this sequence (b=8), A031076 (b=9), A007376 and A033307 (b=10). - Jason Kimberley, Dec 06 2012

[0]cat &cat[Reverse(IntegerToSequence(n,8)):n in[1..31]]; // Jason Kimberley, Dec 07 2012

Flatten[ IntegerDigits[ Range[0, 40], 8]] (* 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[#, 8] &, 105, 0] (* Robert G. Wilson v, Jun 29 2014 *)

from itertools import count, chain, islice
from sympy.ntheory.factor_ import digits
def A054634_gen(): return chain.from_iterable(digits(m, 8)[1:] for m in count(0))
A054634_list = list(islice(A054634_gen(), 30)) # Chai Wah Wu, Jan 07 2022

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

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

A055144 The first n digits of the juxtaposition of the base-3 numbers converted to decimal.

Original entry on oeis.org

1, 5, 16, 48, 145, 436, 1309, 3929, 11789, 35367, 106103, 318310, 954932, 2864798, 8594395, 25783185, 77349555, 232048666, 696145998, 2088437995, 6265313986, 18795941958, 56387825876, 169163477629, 507490432888, 1522471298664

Offset: 1

Patrick De Geest, Apr 15 2000

			1 (1); 12 (5); 121 (16); 1210 (48); 12101 (145); 121011 (436); ...

Cf. A054635, A054637, A055072, A055073, A055143-A055150.

cp's OEIS Frontend

A003137 Write n in base 3 and juxtapose.

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A030373 Write n in base 4 and juxtapose.

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A031076 Write n in base 9 and juxtapose.

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A030998 Write n in base 7 and juxtapose.

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A031035 Write n in base 8 and juxtapose.

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A054634 Champernowne sequence: write n in base 8 and juxtapose.

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

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