A033307 -id:A033307 - cp's OEIS frontend

A244816 The hexagonal spiral of Champernowne, read along the South (or 180-degree) ray.

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A244806, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244817, A244818.

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]]]; f[n_] := 12n^2 - 19n + 8 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(12n^2 - 19n + 8)th almost natural number (A033307), Also see formula section of A056105.

A244817 The hexagonal spiral of Champernowne, read along the 150-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			see A244807 example section for its diagram.

Cf. A007376, A003215, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244818.

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]]]; f[n_] := 3n^2 - 3n + 1 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(3n^2 - 3n + 1)th almost natural number (A033307), Also see formula section of A056105.

A244818 The hexagonal spiral of Champernowne, read along the 120-degree ray.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Jul 06 2014

			See A244807 example section for its diagram.

Cf. A007376, A033577, A244807, A244808, A244809, A244810, A244811, A244812, A244813, A244814, A244815, A244816, A244817.

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]]]; f[n_] := 12n^2 - 17n + 6 (* see formula section of A244807 *); Array[ almostNatural[ f@#, 10] &, 105]

(12n^2-17n+6)th almost natural number (A033307), also see formula section of A056105.

A176942 Champernowne primes.

Original entry on oeis.org

1234567891, 12345678910111, 123456789101112131415161

Offset: 1

Marco Ripà, Jan 27 2011

Primes formed from an initial portion 1234... of the infinite string 12345678910111213... of the concatenation of all positive integers (decimal digits of the Champernowne constant).
From Eric W. Weisstein, Jul 15 2013: (Start)
The next terms are too big to display:
a(4) = 123456789...1121131141 (235 digits)
a(5) = 123456789...6896997097 (2804 digits)
a(6) = 12345...13611362136313 (4347 digits)
a(7) = 123456789...9709971097 (37735 digits)
a(8) has more than 37800 digits. (End)
a(8) has more than 140000 digits. - Tyler Busby, Feb 12 2023

R. W. Stephan, Factors and primes in two Smarandache sequences.

Eric W. Weisstein, Table of n, a(n) for n = 1..6
Eric W. Weisstein, Table of n, a(n) for n = 1..7 (an a-file)
Eric Weisstein's World of Mathematics, Champernowne Constant
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. A007908, A058183, A073175, A053546.
Cf. A007376 (infinite Barbier word = almost-natural numbers: write n in base 10 and juxtapose digits).
Cf. A033307 (decimal expansion of Champernowne constant).
Cf. A071620 (number of digits in the n-th Champernowne prime).
See A265043 for where to end the string of numbers that are being concatenated in order to get the n-th prime.

With[{no=500},FromDigits/@Select[Table[Take[Flatten[IntegerDigits/@Range[no]],n],{n,no}],PrimeQ[FromDigits[#]]&]]  (* Harvey P. Dale, Feb 06 2011 *)
Select[Table[Floor[N[ChampernowneNumber[10], n]*10^n], {n, 24}], PrimeQ] (* Arkadiusz Wesolowski, May 10 2012 *)

A351753 Take the first n digits on the binary Champernowne string (cf. A030302); a(n) gives the starting index of the second occurrence of this n-digit string within the binary Champernowne string.

Original entry on oeis.org

2, 4, 5, 12, 12, 12, 213, 517, 517, 517, 517, 517, 517, 517, 517, 517, 14457, 189569, 258049, 258049, 14144865, 14144865, 14144865, 131391133, 131391133, 199844657, 199844657, 199844657, 1196986333, 1196986333, 5176897753, 5176897753, 5176897753, 5176897753

Offset: 1

Scott R. Shannon, Feb 18 2022

The twenty-first n-digit string is '110111001011101111000' (1808238 decimal) which cannot be readily split into consecutive smaller values implying it is likely its next occurrence is in its natural position, i.e., a(21) = 35876058.

			The binary Champernowne string starts 110111001011101111000100110101011....
a(1) = 2 as the second occurrence of '1' within the string starts at index 2.
a(2) = 4 as the second occurrence of '11' within the string starts at index 4.
a(3) = 5 as the second occurrence of '110' within the string starts at index 5.
a(4) = 12 as the second occurrence of '1101' within the string starts at index 12.

Rémy Sigrist, Table of n, a(n) for n = 1..43
Rémy Sigrist, C++ program

Cf. A030302, A030303, A033307, A337227.

from itertools import count
def A351753(n):
    s1, s2 = tuple(), tuple()
    for i, s in enumerate(int(d) for n in count(1) for d in bin(n)[2:]):
        if i < n:
            s1 += (s,)
            s2 += (s,)
        else:
            s2 = s2[1:]+(s,)
            if s1 == s2:
                return i-n+2 # Chai Wah Wu, Feb 18 2022
(C++) // See Links section.

a(18)-a(20) corrected and a(21)-a(34) added by Chai Wah Wu, Feb 18 2022

A065649 Permutation of nonnegative integers based on Champernowne's constant 0.123456789101112131415...

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 21, 31, 41, 12, 51, 13, 61, 14, 71, 15, 81, 16, 91, 17, 101, 18, 111, 19, 22, 20, 32, 121, 42, 52, 62, 23, 72, 24, 82, 25, 92, 26, 102, 27, 112, 28, 122, 29, 33, 30, 43, 131, 53, 132, 63, 73, 83, 34, 93, 35, 103, 36, 113, 37, 123, 38

Offset: 0

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

A261293(n) = a(a(n)). - Reinhard Zumkeller, Aug 14 2015

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Klaus Strassburger, Illustration
Index entries for sequences that are permutations of the natural numbers

A033307, A065648, A065650.

Cf. A261279 (fixed points), A261293, A261333.

a065649 n = a065649_list !! n
a065649_list = zipWith (+)
               (map ((* 10) . subtract 1) a065648_list) (0 : a033307_list)
-- Reinhard Zumkeller, Aug 13 2015

a(n) = if n = 0 then 0 else 10*(A065648(n)-1) + A033307(n-1).

Offset and defining formula adjusted by Reinhard Zumkeller, Aug 13 2015

A065650 Inverse permutation to A065649.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 15, 17, 19, 21, 23, 25, 27, 29, 31, 12, 30, 37, 39, 41, 43, 45, 47, 49, 51, 13, 32, 50, 59, 61, 63, 65, 67, 69, 71, 14, 34, 52, 70, 81, 83, 85, 87, 89, 91, 16, 35, 54, 72, 90, 103, 105, 107, 109, 111, 18, 36, 56, 74, 92, 110, 125, 127

Offset: 0

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

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A033307, A065648, A065649, A261334.

Cf. A261279 (fixed points), A261294.

import Data.List (elemIndex); import Data.Maybe (fromJust)
a065650 = fromJust . (`elemIndex` a065649_list)
-- Reinhard Zumkeller, Aug 13 2015

Offset changed by Reinhard Zumkeller, Aug 13 2015

A073175 First occurrence of an n-digit prime as a substring in the concatenation of the natural numbers 12345678910111213141516171819202122232425262728293031....

Original entry on oeis.org

2, 23, 101, 4567, 67891, 789101, 4567891, 23456789, 728293031, 1234567891, 45678910111, 678910111213, 1222324252627, 12345678910111, 415161718192021, 3637383940414243, 12223242526272829, 910111213141516171

Offset: 1

Zak Seidov, Aug 22 2002

This is to Champernowne's constant 0.12345678910111213... (Sloane's A033307) as A073062 is to A033308 Decimal expansion of Copeland-Erdos constant: concatenate primes. - Jonathan Vos Post, Aug 25 2008

			Take 1234567891011121314151617....; a(4)=4567 because the first 4-digit prime in the sequence is 4567.
1213 is < 4567 but occurs later in the string.
a(5) = 67891 is the first occurrence of a five-digit substring that is a prime, 12345(67891)011121314...
a(1) = 2 = prime(1). a(2) = 23 = prime(9). a(3) = 571 = prime(105). a(4) = 2357 = prime(350). a(5) = 11131 = prime(1349). - _Jonathan Vos Post_, Aug 25 2008

Robert Israel, Table of n, a(n) for n = 1..999
Eric W. Weisstein, Champernowne Constant. [From _Jonathan Vos Post_, Aug 25 2008]
Eric W. Weisstein, Copeland-Erdos Constant. [From _Jonathan Vos Post_, Aug 25 2008]

Cf. A003617. - M. F. Hasler, Aug 23 2008

Cf. A000040, A033307, A033308, A073062. - Jonathan Vos Post, Aug 25 2008

N:= 1000: # to use the concatenation of 1 to N
L:= NULL:
for n from 1 to N do
  L:= L, op(ListTools:-Reverse(convert(n,base,10)))
od:
L:= [L]:
nL:= nops(L);
f:= proc(n) local k,B,x;
  for k from 1 to nL-n+1 do
    B:= L[k..k+n-1];
    x:= add(B[i]*10^(n-i),i=1..n);
    if isprime(x) then return x fi
  od;
false;
end proc:
seq(f(n),n=1..100); # Robert Israel, Aug 16 2018

p200=Flatten[IntegerDigits[Range[200]]]; Do[pn=Partition[p200, n, 1]; ln=Length[pn]; tab=Table[Sum[10^(n-k)*pn[[i, k]], {k, n}], {i, ln}]; Print[{n, Select[tab, PrimeQ][[1]]}], {n, 20}]

{s=Vec(Str(c=1)); for(d=1,30, for(j=1,9e9,
#sM. F. Hasler, Aug 23 2008

Edited by N. J. A. Sloane, Aug 19 2008 at the suggestion of R. J. Mathar

A340207 Constant whose decimal expansion is the concatenation of the largest n-digit square A061433(n), for n = 1, 2, 3, ...

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Jan 01 2021

The terms of sequence A339978 converge to this sequence of digits, and to this constant, up to powers of 10.

			The largest square with 1, 2, 3, 4, ... digits is, respectively, 9 = 3^2, 81 = 9^2, 961 = 31^2, 9801 = 99^2, ....
Here we list the sequence of digits of these numbers: 9; 8, 1; 9, 6, 1; 9, 8, 0, 1; 9, 9, 8, 5, 6; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.98196198...

Harvey P. Dale, Table of n, a(n) for n = 0..1000

Cf. A061433 (largest n-digit square), A339978 (has this as "limit"), A340208 (same with "smallest n-digit cube", limit of A215692), A340209 (same for cubes, limit of A340115), A340220 (same for primes), A340219 (similar, with smallest primes, limit of A215641), A340222 (same for semiprimes), A340221 (same for smallest semiprimes, limit of A215647).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

lnds[k_]:=Module[{c=Sqrt[10^k]},If[IntegerQ[c],(c-1)^2,Floor[c]^2]]; Flatten[IntegerDigits/@(lnds/@Range[15])] (* Harvey P. Dale, Dec 16 2021 *)

concat([digits(sqrtint(10^k-1)^2)|k<-[1..14]]) \\ as seq. of digits
c(N=20)=sum(k=1,N,.1^(k*(k+1)/2)*sqrtint(10^k-1)^2) \\ as constant

c = 0.9819619801998569980019998244999800019999508849999800001999995155...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*floor(10^(k/2)-1)^2
a(-n(n+1)/2) = 9 for all n >= 2.

A340208 Constant whose decimal expansion is the concatenation of the smallest n-digit cube A061434(n), for n = 1, 2, 3, ...

Original entry on oeis.org

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

Offset: 0

M. F. Hasler, Dec 31 2020

Every third smallest n-digit cube (i.e., for n = 3k + 1, k >= 0) is 10^k, which explains the chunks of (1,0,...,0), cf. formula.

The terms of sequence A215692 converge to this sequence of digits, and to this constant, up to powers of 10.

			The smallest cube with 1, 2, 3, 4, ... digits is, respectively, 1, 27 = 3^3, 125 = 5^3, 1000 = 10^3, .... Here we list the sequence of digits of these numbers: 1; 2, 7; 1, 2, 5; 1, 0, 0, 0; ...
This can be considered, as for the Champernowne and Copeland-Erdős constants, as the decimal expansion of a real constant 0.1271251000106481...
As a triangle, in which row n contains the decimal expansion of the smallest n-digit cube:
  1
  2 7
  1 2 5
  1 0 0 0
  1 0 6 4 8
  1 0 3 8 2 3
  1 0 0 0 0 0 0
  1 0 0 7 7 6 9 6
  ...

Zhuorui He, Table of n, a(n) for n = 0..11324 (first 150 rows)

Cf. A061434 (smallest n-digit cube), A215692 (has this as "limit"), A340209 (same with largest n-digit cubes, limit of A340115), A340206 (same for squares, limit of A215689), A340219 (same for primes, limit of A215641), A340221 (same for semiprimes, limit of A215647).

Cf. A033307 (Champernowne constant), A030190 (binary), A001191 (concatenation of all squares), A134724 (cubes), A033308 (primes: Copeland-Erdős constant).

concat([digits(ceil(10^((k-1)/3))^3)|k<-[1..14]]) \\ as seq. of digits
c(N=12)=sum(k=1,N,.1^(k*(k+1)/2)*ceil(10^((k-1)/2))^2) \\ as constant

c = 0.12712510001064810382310000001007769610054462510000000001000787387510002657...
  = Sum_{k >= 1} 10^(-k(k+1)/2)*ceiling(10^((k-1)/3))^2
a(-n(n+1)/2) = 1 for all n >= 2;
a(k) = 0 for -3n(3n+1)/2 > k > -(3n+1)(3n+2)/2, n >= 0.

cp's OEIS Frontend

A244816 The hexagonal spiral of Champernowne, read along the South (or 180-degree) ray.

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A244817 The hexagonal spiral of Champernowne, read along the 150-degree ray.

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A244818 The hexagonal spiral of Champernowne, read along the 120-degree ray.

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A176942 Champernowne primes.

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A351753 Take the first n digits on the binary Champernowne string (cf. A030302); a(n) gives the starting index of the second occurrence of this n-digit string within the binary Champernowne string.

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A065649 Permutation of nonnegative integers based on Champernowne's constant 0.123456789101112131415...

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A065650 Inverse permutation to A065649.

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A073175 First occurrence of an n-digit prime as a substring in the concatenation of the natural numbers 12345678910111213141516171819202122232425262728293031....

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