A033307 -id:A033307 - cp's OEIS frontend

A337227 a(n) = difference between the starting positions of the first two occurrences of n in the Champernowne string (starting at 0) 01234567891011121314151617181920... (cf. A033307).

11, 9, 13, 14, 15, 16, 17, 18, 19, 20, 180, 1, 19, 37, 55, 73, 91, 109, 127, 145, 221, 166, 1, 19, 37, 55, 73, 91, 109, 127, 231, 149, 233, 1, 19, 37, 55, 73, 91, 109, 241, 132, 243, 244, 1, 19, 37, 55, 73, 91, 251, 115, 253, 254, 255, 1, 19, 37, 55, 73, 261, 98, 263, 264, 265, 266, 1, 19

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Aug 19 2020

Consider the infinite string
   01234567891011121314151617181920... (cf. A033307)
formed by the concatenation of all decimal digits of all nonnegative numbers. From the position of the first digit of the first occurrence of the number n find the number of digits one has to move forward to get to the start of the second occurrence of n. This is a(n).

			The infinite string corresponding to the concatenation of all decimal digits >=0 starts "012345678910111213141516171819202122232425....".
a(0) = 11 because '0' appears at positions 1 and 12.
a(1) = 9 because '1' appears at positions 2 and 11.
a(10) = 180 because '10' appears starting at positions 11 and 191.
a(11) = 1 because '11' appears starting at positions 13 and 14.

Scott R. Shannon, Image of the first 100000 terms.

Cf. A007376, A331162, A033307.

Cf. A342162.

from itertools import count
def A337227(n):
    s1 = tuple(int(d) for d in str(n))
    s2 = s1
    for i, s in enumerate(int(d) for k in count(n+1) for d in str(k)):
        s2 = s2[1:]+(s,)
        if s2 == s1:
            return i+1 # Chai Wah Wu, Feb 18 2022

A065648 a(0) = 1 and for n > 0: a(n) = number of indices 0 <= i <= n with A033307(i)=A033307(n), where A033307 is the sequence of decimal digits of Champernowne's constant 0.123456789101112131415...

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 2, 6, 2, 7, 2, 8, 2, 9, 2, 10, 2, 11, 2, 12, 2, 3, 3, 4, 13, 5, 6, 7, 3, 8, 3, 9, 3, 10, 3, 11, 3, 12, 3, 13, 3, 4, 4, 5, 14, 6, 14, 7, 8, 9, 4, 10, 4, 11, 4, 12, 4, 13, 4, 14, 4, 5, 5, 6, 15, 7, 15, 8, 15, 9, 10, 11, 5, 12, 5, 13, 5, 14, 5, 15, 5, 6, 6

Offset: 0

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Nov 09 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A033307, A065649, A065650.

Cf. A256100.

a065648 n = a065648_list !! n
a065648_list = f (0 : a033307_list) $ take 10 $ repeat 1 where
   f (d:ds) counts = y : f ds (xs ++ (y + 1) : ys) where
                           (xs, y:ys) = splitAt d counts
-- Reinhard Zumkeller, Aug 13 2015

Definition and offset adjusted to follow a later change of A033307. - Reinhard Zumkeller, Aug 13 2015

A071620 Integer lengths of the Champernowne primes (concatenation of first a(n) entries (digits) of A033307 is prime).

Original entry on oeis.org

10, 14, 24, 235, 2804, 4347, 37735

Offset: 1

Robert G. Wilson v, Jun 21 2002

Next term has n > 113821. - Eric W. Weisstein, Nov 04 2015

Also: concatenation of A007376(1 .. a(n)) is prime. - M. F. Hasler, Oct 23 2019

Eric Weisstein's World of Mathematics, Champernowne Constant Digits
Eric Weisstein's World of Mathematics, Consecutive Number Sequences
Eric Weisstein's World of Mathematics, Constant Primes
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Smarandache Prime

Cf. A007376 (infinite Barbier word = almost-natural numbers: write n in base 10 and juxtapose digits).
Cf. A033307 (decimal expansion of Champernowne constant), A176942 (the corresponding primes of length a(n)), A265043.
Cf. A072125.

f[0] = 0; f[n_Integer] := 10^(Floor[Log[10, n]] + 1)*f[n - 1] + n; Do[If[PrimeQ[FromDigits[Take[IntegerDigits[f[n]], n]]], Print[n]], {n, 1, 3000}]
Cases[FromDigits /@ Rest[FoldList[Append, {}, RealDigits[N[ChampernowneNumber[], 1000]][[1]]]],  p_?PrimeQ :> IntegerLength[p]] (* Eric W. Weisstein, Nov 04 2015 *)

from itertools import count, islice
from sympy import isprime
def A071620_gen(): # generator of terms
    c, l = 0, 0
    for n in count(1):
        for d in str(n):
            c = 10*c+int(d)
            l += 1
            if isprime(c):
                yield l
A071620_list = list(islice(A071620_gen(),5)) # Chai Wah Wu, Feb 27 2023

Edited by Charles R Greathouse IV, Apr 28 2010
a(6) = 4347 from Eric W. Weisstein, Jul 14 2013
a(7) = 37735 from Eric W. Weisstein, Jul 15 2013

A086044 Position of first occurrence of n in concatenated numbers (A033307, Champernowne constant).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 2, 17, 19, 21, 23, 25, 27, 29, 31, 16, 35, 3, 39, 41, 43, 45, 47, 49, 51, 18, 38, 57, 4, 61, 63, 65, 67, 69, 71, 20, 40, 60, 79, 5, 83, 85, 87, 89, 91, 22, 42, 62, 82, 101, 6, 105, 107, 109, 111, 24, 44, 64, 84, 104, 123, 7, 127

Offset: 0

Reinhard Zumkeller, Jul 07 2003

A033307(a(n)) = A000030(n).

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

Cf. A086045, A086048.

With[{ch=Flatten[IntegerDigits/@Range[0,500]]},Table[ SequencePosition[ ch,IntegerDigits[n],1][[1]],{n,0,70}]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 07 2020 *)

A252043 a(n) is the concatenation of first n terms of A033307.

Original entry on oeis.org

1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567891, 12345678910, 123456789101, 1234567891011, 12345678910111, 123456789101112, 1234567891011121, 12345678910111213, 123456789101112131, 1234567891011121314

Offset: 1

José de Jesús Camacho Medina, Dec 15 2014

			a(3)=123.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A007908 (concatenate 1 through n), A033307.

a[0]:= 0;
count:= 0:
for x from 1 to 30 do
  L:= convert(x,base,10);
  for i from 1 to nops(L) do
    count:= count+1;
    a[count]:= a[count-1]*10+L[-i];
  od
od:
seq(a[i],i=1..count); # Robert Israel, Jan 11 2015

b[1] = 1
b[n_] := b[n - 1]*10^(Floor[Log[10, 10n]]) + n
Table[Floor[b[n] /10^(n)], {n, 10, 200}]
Module[{nn=20,ch},ch=RealDigits[ChampernowneNumber[],10,nn][[1]];Table[ FromDigits[ Take[ch,n]],{n,nn}]] (* Harvey P. Dale, Aug 31 2015 *)

from itertools import islice
def bgen(): yield from (c for n in count(1) for c in str(n) )
def agen():
    s, g = "", bgen()
    while True:
        s += next(g); yield int(s)
print(list(islice(agen(), 20))) # Michael S. Branicky, Oct 25 2022

a(n) = floor(C*10^n) with C the Champernowne constant, 0.123456789101112131415..., A033307.

a(n) = floor(A007908(n)/10^n) For n>=10.

Definition corrected by Zak Seidov, Jan 18 2015

A058068 Denominators of convergents to Champernowne constant (A033307).

Original entry on oeis.org

1, 8, 73, 81, 12075796, 12075877, 24151673, 36227550, 169061873, 205289423, 374351296, 579640719, 2113273453, 9032734531, 11146007984, 20178742515, 31324750499, 490050000000

Offset: 0

N. J. A. Sloane, Nov 24 2000

a(18) has 178 digits and is too large to include here. - Lucas A. Brown, Oct 22 2022

Robert Israel, Table of n, a(n) for n = 0..39
Lucas A. Brown, A058068+9.py.
E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.

Cf. A033307, A058069.

Digits := 100: E := A033307; convert(evalf(E),confrac,50,'cvgts'): cvgts;

More terms from Robert Israel, Jan 06 2019

A058069 Numerators of convergents to Champernowne constant (A033307).

Original entry on oeis.org

0, 1, 9, 10, 1490839, 1490849, 2981688, 4472537, 20871836, 25344373, 46216209, 71560582, 260897955, 1115152402, 1376050357, 2491202759, 3867253116, 60499999499

Offset: 0

N. J. A. Sloane, Nov 24 2000

a(18) has 177 digits and is too large to include here. - Lucas A. Brown, Sep 02 2022

Lucas A. Brown, Table of n, a(n) for n = 0..39
Lucas A. Brown, A058068+9.py.
E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.

Cf. A033307, A058068.

Digits := 100: E := A033307; convert(evalf(E),confrac,50,'cvgts'): cvgts;

Module[{champ=FromDigits[Flatten[IntegerDigits/@Range[20]]]}, Numerator[ Convergents[ champ/10^IntegerLength[champ]]]] (* Harvey P. Dale, Dec 17 2014 *)

a(17) from Lucas A. Brown, Sep 02 2022

A086045 Position of second occurrence of n in concatenated numbers (A033307, Champernowne constant).

Original entry on oeis.org

12, 11, 16, 18, 20, 22, 24, 26, 28, 30, 191, 14, 15, 54, 74, 94, 114, 134, 154, 174, 252, 33, 36, 37, 76, 96, 116, 136, 156, 176, 282, 53, 55, 58, 59, 98, 118, 138, 158, 178, 312, 73, 75, 77, 80, 81, 120, 140, 160, 180, 342, 93, 95, 97, 99, 102, 103, 142

Offset: 0

Reinhard Zumkeller, Jul 07 2003

A033307(a(n)) = A000030(n); a(n)>A086044(n);

A033307(a(n)+i) = A033307(A086044(n)+i), 0<=i < A055642(n).

Paul Tek, Table of n, a(n) for n = 0..10000

Join[{12},Module[{nn=350,cno},cno=Flatten[IntegerDigits/@Range[nn]];Table[ SequencePosition[ cno,IntegerDigits[n],2][[2,1]],{n,60}]]+1] (* Harvey P. Dale, Dec 09 2022 *)

A098732 Consider the sequence {b(n), n >= 1} of digits of the integers: 0 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... (A033307); a(n) = n - b(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 12, 12, 13, 14, 14, 16, 15, 18, 16, 20, 17, 22, 18, 24, 19, 26, 20, 28, 21, 29, 32, 31, 33, 33, 34, 35, 35, 37, 36, 39, 37, 41, 38, 43, 39, 45, 40, 47, 41, 48, 52, 50, 53, 52, 54, 54, 55, 56, 56, 58, 57, 60, 58, 62, 59, 64, 60, 66, 61, 67, 72, 69

Offset: 1

Alexandre Wajnberg, Sep 30 2004

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A098727, etc.

First[#]-Last[#]&/@Module[{ch=Flatten[IntegerDigits/@Range[ 0,50]]},Thread[ {Range[Length[ch]],ch}]] (* Harvey P. Dale, Oct 15 2014 *)

More terms from Joshua Zucker, May 18 2006

A228584 Start with decimal expansion of Champernowne constant (A033307) and repeatedly remove the first digit between two neighbors (after the decimal point) having the same parity.

Original entry on oeis.org

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

Offset: 0

Alois P. Heinz, Aug 26 2013

Parity of digits is 1, 1, 0, 0, ... = A133872.

			Start with A033307 (decimal expansion of Champernowne's constant): 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,...
Now erase a digit when it is placed between two digits having the same parity -- and do this repeatedly.
Example:
1,2,3,... erase 2
You get now:
1,3,4,5,... erase 4
You get now: 1,3,5,... erase 3
You get now: 1,5,6,7,... erase 6
You get now: 1,5,7,... erase 5 etc.
The surviving digits are this sequence:
1,9,2,2,9,9,4,4,9,9,6,6,9,9,8,8,9,1,0,0,1,1,0,...
and the original "untouched" positive integers, A228585:
1, 29, 49, 69, 89, 219, 239, 259, 279, 419, 439, ...
We obtain a new constant,
0.1922994499669988910011003100510071...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Eric Angelini et al., Champernowne sieved and follow-up messages on the SeqFan list, Aug 26 2013

Cf. A033307, A133872, A228585 (untouched numbers in the removal process).

cp's OEIS Frontend

A337227 a(n) = difference between the starting positions of the first two occurrences of n in the Champernowne string (starting at 0) 01234567891011121314151617181920... (cf. A033307).

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A065648 a(0) = 1 and for n > 0: a(n) = number of indices 0 <= i <= n with A033307(i)=A033307(n), where A033307 is the sequence of decimal digits of Champernowne's constant 0.123456789101112131415...

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A071620 Integer lengths of the Champernowne primes (concatenation of first a(n) entries (digits) of A033307 is prime).

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A086044 Position of first occurrence of n in concatenated numbers (A033307, Champernowne constant).

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A252043 a(n) is the concatenation of first n terms of A033307.

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Maple

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A058068 Denominators of convergents to Champernowne constant (A033307).

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A058069 Numerators of convergents to Champernowne constant (A033307).

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Maple

Mathematica

Extensions

A086045 Position of second occurrence of n in concatenated numbers (A033307, Champernowne constant).

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