A033308 -id:A033308 - cp's OEIS frontend

A097487 Write the nonprime positive integers on labels in numerical order, forming an infinite sequence L. Now consider the succession of single digits of A000040 (prime numbers): 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 3 7 4 1 4 3 4 7 5 3 ... (A033308). This sequence gives an arrangement L that produces the same succession of digits, subject to the constraint that the smallest unused label must be used that does not lead to a contradiction.

235, 711, 1, 3171, 9, 232, 93, 1374, 14, 34, 75, 35, 96, 16, 77, 1737, 98, 38, 99, 710, 110, 310, 71091, 1312, 713, 1137, 1391, 4, 91, 51, 15, 716, 316, 717, 3179, 18, 119, 11931, 97199, 21, 12, 2322, 72, 292, 33, 2392, 412, 512, 57, 26, 32, 6, 92, 712, 772, 8

Offset: 1

Eric Angelini, Sep 19 2004; corrected Sep 23 2004

This could be roughly rephrased like this: "Rewrite in the most economical way the prime numbers 'pattern' using only nonprime numbers. Do not use any nonprime twice."

			We must begin with 2,3,5,7,11,13,... and we cannot represent "2" with the label "2" or "23", so the next possibility is the label "235" (first available nonprime number in L).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Angelini, Jeux de suites, in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

Cf. A068663, A097968, A098067, A098099.

f[lst_List, k_] := Block[{L = lst, g, a = {}, m = 0}, g[] := {Set[m, First@ FromDigits@ Append[IntegerDigits@ m, First@ #]], Set[L, Last@ #]} &@ TakeDrop[L, 1]; Do[g[]; While[Or[PrimeQ@ m, MemberQ[a, m]], g[]]; AppendTo[a, m]; m = 0, {k}]; a]; f[Flatten@ Map[IntegerDigits, Prime@ Range@ 200], 56] (* Michael De Vlieger, Nov 29 2015, Version 10.2 *)

A086048 Position of first occurrence of n in concatenated primes (A033308, Copeland-Erdos constant).

Original entry on oeis.org

1, 6, 2, 3, 22, 4, 32, 5, 42, 13, 48, 6, 63, 8, 23, 78, 33, 10, 96, 12, 590, 111, 114, 2, 129, 132, 138, 64, 150, 16, 159, 9, 15, 124, 25, 3, 189, 20, 43, 73, 207, 22, 216, 24, 228, 234, 237, 26, 249, 76, 258, 79, 264, 28, 270, 276, 279, 4, 291, 30, 300, 32, 830

Offset: 1

Reinhard Zumkeller, Jul 07 2003

A033308(a(n))=A000030(n);

A033308(a(n)+i)=A033307(A086044(n)+i), 0<=i < A055642(n).

Eric Weisstein's World of Mathematics, Copeland-Erdos Constant

Cf. A086049, A086044.

A086049 Position of first occurrence of n-th prime in concatenated primes (A033308, Copeland-Erdos constant).

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 10, 12, 2, 16, 9, 20, 22, 24, 26, 28, 30, 32, 34, 5, 38, 40, 42, 44, 46, 48, 51, 54, 57, 7, 63, 8, 19, 72, 75, 78, 81, 84, 33, 37, 93, 96, 99, 102, 105, 108, 111, 114, 117, 120, 123, 126, 129, 132, 135, 138, 141, 64, 147, 150, 153, 16, 159, 67, 18

Offset: 1

Reinhard Zumkeller, Jul 07 2003

A033308(a(n)) = A000030(A000040(n));

A033308(a(n)+i) = A033307(A086044(A000040(n))+i), 0 <= i < A055642(A000040(n)).

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Copeland-Erdos Constant

Cf. A033307, A033308, A086044, A086048, A086050.

N:= 100: # for a(1)..a(N)
S:= cat(".",seq(ithprime(i),i=1..N)):
seq(StringTools:-Search(convert(ithprime(i),string),S),i=1..N); # Robert Israel, Jun 28 2019

A165449 Write the prime numbers in a string: 2357111317192329... (cf. A033308). The sequence gives the first position in the string for natural numbers.

Original entry on oeis.org

5, 1, 2, 21, 3, 31, 4, 41, 12, 47, 5, 62, 7, 22, 77, 32, 9, 95, 11, 589, 110, 113, 1, 128, 131, 137, 63, 149, 15, 158, 8, 14, 123, 24, 2, 188, 19, 42, 72, 206, 21, 215, 23, 227, 233, 236, 25, 248, 75, 257, 78, 263, 27, 269, 275, 278, 3, 290, 29, 299, 31, 829

Offset: 1

Rémy Sigrist, Sep 20 2009

Same as A229190 but omitting the a(0) term.

Defined for all a by the normality of the Copeland-Erdős constant. - Aaron Weiner, Sep 19 2013

			The first occurrence of "111" in the string is 5, so a(111)=5.

Nathaniel Johnston, Table of n, a(n) for n = 1..5000

Cf. A229190 (same sequence but including the a(0) term).

Cf. A031297.

with(StringTools): s:="": for n from 1 to 300 do s:=cat(s,convert(ithprime(n),string)): od: seq(Search(convert(n,string),s),n=1..62); # Nathaniel Johnston, May 26 2011

With[{prd=Flatten[IntegerDigits/@Prime[Range[1000]]],nn=10},Flatten[ Table[ SequencePosition[ prd,IntegerDigits[ n],1],{n,70}],1]][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, May 12 2019 *)

from sympy import primerange
from itertools import count, takewhile
def afind(plimit):
  s = "".join(str(p) for p in primerange(1, plimit+1))
  return [1+s.find(str(n)) for n in takewhile(lambda i: str(i) in s, count(1))]
print(afind(10**4)) # Michael S. Branicky, May 01 2021

A227529 Copeland-Erdős constant primes (concatenation of entries (digits) of A033308 is prime).

Original entry on oeis.org

2, 23, 2357, 23571113171

Offset: 1

Eric W. Weisstein, Jul 14 2013

Primes obtained by taking consecutive decimal digits (starting with the first) of the Copeland-Erdős constant.
The next terms are too big to display: a(5) = 235711131...6917017097 (353 digits), a(6) = 235711131...1701709719 (355 digits), ...
See A227530 for an equivalent but more compact way of listing the terms, namely, by giving the number of digits of the constant A033308 that must be taken to get a prime. - M. F. Hasler, Apr 24 2017

Eric W. Weisstein, Table of n, a(n) for n = 1..7
Hans Havermann and Eric W. Weisstein (first 11 terms), Table of n, a(n) for n = 1..14
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant

Cf. A227530 (number of decimal digits in the n-th Copeland-Erdős prime).
Cf. A019518, A069151.
Cf. A033308 (Decimal expansion of Copeland-Erdős constant: concatenate primes).

A098103 Consider the succession of single digits of the primes (A000040): 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 ... (A033308). This sequence is the lexicographically earliest derangement of A000040 that produces the same succession of digits.

Original entry on oeis.org

23, 5, 7, 11, 13, 17, 19, 2, 3, 293, 137, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 1371391491511, 571, 631, 67173179181191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Eric Angelini, Sep 22 2004

Derangement here means a(n) != A000040(n) for all n.
Original name: "Write each prime number >0 on a single label. Put the labels in numerical order to form an infinite sequence L. Consider the succession of single digits of L: 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 3 7 4 1 4 3 4 7 5 3 5 9 6 1 6 7 7 1 7 3 7 9... (see A033308). The sequence S gives a rearrangement of the labels that reproduces the same succession of digits, subject to the constraints that a label of L cannot represent itself, and the smallest label must be used that does not lead to a contradiction."
This could be roughly rephrased like this: "Rewrite in the most economical way the 'prime numbers pattern' using only prime numbers, but rearranged. Do not use any prime more than once."
a(180) has over 1000 digits. - Danny Rorabaugh, Nov 29 2015

			We must begin with "2,3,5,7,11,..." and we cannot have the first term be 2, the first prime, so the smallest available prime is 23.

Danny Rorabaugh, Table of n, a(n) for n = 1..179
Eric Angelini, Sequence re-writing
Eric Angelini, Sequence re-writing [Cached copy, with permission]

For other sequences of this type, cf. A098067.

f[lst_List, k_] := Block[{L = lst, g, a = {}, m = 0}, g[] := {Set[m, First@ FromDigits@ Append[IntegerDigits@ m, First@ #]], Set[L, Last@ #]} &@ TakeDrop[L, 1]; Do[g[]; While[Or[m == Prime[Length@ a + 1], ! PrimeQ@ m, MemberQ[a, m]], g[]]; AppendTo[a, m]; m = 0, {k}]; a]; f[Flatten@ Map[IntegerDigits, Prime@ Range@ 120], 53] (* Michael De Vlieger, Nov 29 2015, Version 10.2 *)

def A098103(n):
  Pr, p, s, A, i = Primes(), 2, "", [], 1
  while len(A)A098103(179) # Danny Rorabaugh, Nov 29 2015

Name, Comments, and Example edited by Danny Rorabaugh, Nov 28 2015

Corrected and extended by Danny Rorabaugh, Nov 29 2015

A165450 Sequence gives prime numbers which occur in A033308 ahead of their natural place.

Original entry on oeis.org

23, 31, 71, 113, 131, 137, 167, 173, 271, 293, 311, 313, 317, 331, 347, 359, 373, 379, 389, 491, 571, 593, 631, 673, 677, 719, 733, 761, 773, 811, 877, 911, 941, 971, 977, 983, 997, 1031, 1091, 1103, 1109, 1171, 1193, 1223, 1231, 1277, 1283, 1291, 1361

Offset: 1

Rémy Sigrist, Sep 20 2009

Same idea as A116700 applied to prime numbers.

A227530 Integer lengths of the n-th Copeland-Erdős prime (concatenation of the first n entries (digits) of A033308 is prime).

Original entry on oeis.org

1, 2, 4, 11, 353, 355, 499, 1171, 1543, 5719, 11048, 68433, 97855, 292447

Offset: 1

Eric W. Weisstein, Jul 14 2013

a(15) > 300000. - Eric W. Weisstein (according to Mark Rodenkirch as of Dec 13 2015)
The larger of these are probable primes.
a(15) > 5*10^5 according to Mark Rodenkirch, June 2016. - M. F. Hasler, Apr 24 2017

Mark Rodenkirch (a.k.a. "rogue"), Copeland-Erdos Constant Primes, MersenneForum.org, June 2016.
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant Digits

Cf. A227529 (primes corresponding to a(n) digits of the Copeland-Erdős constant), A019518, A069151.

Cf. also A033308 (Decimal expansion of Copeland-Erdős constant: concatenate primes).

a(12) = 68433 from Eric W. Weisstein, Aug 17 2013
a(13) = 97855 from Eric W. Weisstein, Oct 24 2015
a(14) = 292447 from Eric W. Weisstein, Dec 17 2015

A086050 Position of second occurrence of n-th prime in concatenated primes (A033308, Copeland-Erdos constant).

Original entry on oeis.org

14, 9, 28, 11, 7, 19, 37, 99, 14, 121, 18, 39, 130, 219, 178, 184, 187, 238, 88, 11, 91, 94, 154, 202, 106, 481, 493, 2534, 529, 60, 146, 61, 69, 689, 254, 562, 287, 570, 87, 90, 1110, 578, 440, 586, 355, 442, 491, 602, 1153, 606, 1181, 1134, 1233, 1142

Offset: 1

Reinhard Zumkeller, Jul 07 2003

A033308(a(n))=A000030(A000040(n)); a(n)>A086049(n);

A033308(a(n)+i)=A033308(A086048(n)+i), 0<=i < A055642(A000040(n)).

A066707 Incrementally largest terms in the continued fraction for the constant given by Sum_{k>=0} A033308(k) / 2^k = 2.89104866587305422....

Original entry on oeis.org

2, 8, 10, 32, 39, 5903, 135598

Offset: 1

Robert G. Wilson v, Jan 14 2002

Previous name: Incrementally largest terms in the continued fraction for Copeland-Erdos constant 0.235711... (concatenate primes).

Appears to be a mistaken version of A033310 in which base 2 has been used in the "concatenation" of the digits in A033308 while still interpreting the digits of A033308 as decimal. - Sean A. Irvine, Nov 03 2023

Cf. A030168, A033310.

a = {}; Do[a = Append[a, IntegerDigits[ Prime[n]]], {n, 1, 5*10^4} ]; b = ContinuedFraction[ N[ FromDigits[{Flatten[a], 0}, 2], 5*10^4]]; c = -1; d = {}; Do[ If[b[[n]] > c, c = b[[n]]; d = Append[d, c]], {n, 1, 48336} ]; d

Name changed to match data by Sean A. Irvine, Nov 03 2023

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A086048 Position of first occurrence of n in concatenated primes (A033308, Copeland-Erdos constant).

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A086049 Position of first occurrence of n-th prime in concatenated primes (A033308, Copeland-Erdos constant).

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A165449 Write the prime numbers in a string: 2357111317192329... (cf. A033308). The sequence gives the first position in the string for natural numbers.

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A227529 Copeland-Erdős constant primes (concatenation of entries (digits) of A033308 is prime).

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A098103 Consider the succession of single digits of the primes (A000040): 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 ... (A033308). This sequence is the lexicographically earliest derangement of A000040 that produces the same succession of digits.

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A165450 Sequence gives prime numbers which occur in A033308 ahead of their natural place.

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A227530 Integer lengths of the n-th Copeland-Erdős prime (concatenation of the first n entries (digits) of A033308 is prime).

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Comments

Links

Crossrefs

Extensions

A086050 Position of second occurrence of n-th prime in concatenated primes (A033308, Copeland-Erdos constant).

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Author

Keywords

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A066707 Incrementally largest terms in the continued fraction for the constant given by Sum_{k>=0} A033308(k) / 2^k = 2.89104866587305422....

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