A074721 -id:A074721 - cp's OEIS frontend

A033308 Decimal expansion of Copeland-Erdős constant: concatenate primes.

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

Offset: 0

Eric W. Weisstein

The number .23571113171923.... was proved normal in base 10 by Copeland and Erdős but is not known to be normal in other bases. - Jeffrey Shallit, Mar 14 2008
Could be read (with indices 1, 2, ...) as irregular table whose n-th row lists the A097944(n) digits of the n-th prime A000040(n). - M. F. Hasler, Oct 25 2019
Named after the American mathematician Arthur Herbert Copeland (1898-1970) and the Hungarian mathematician Paul Erdős (1913-1996). - Amiram Eldar, May 29 2021
This constant is irrational but it is not (yet) known to be transcendental. - Charles R Greathouse IV, Feb 03 2025

			0.235711131719232931374143475359616771737983899710110310710911312...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 6.9, p. 442.
Glyn Harman, One hundred years of normal numbers, in M. A. Bennett et al., eds., Number Theory for the Millennium, II (Urbana, IL, 2000), A K Peters, Natick, MA, 2002, pp. 149-166.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

T. D. Noe, Table of n, a(n) for n = 0..2000
John M. Campbell, The prime-counting Copeland-Erdős constant, arXiv:2309.13520 [math.NT], 2023.
A. H. Copeland and P. Erdős, Note on Normal Numbers, Bull. Amer. Math. Soc., Vol. 52, No. 10 (1946), pp. 857-860.
Mikołaj Morzy, Tomasz Kajdanowicz, and Przemysław Kazienko, On Measuring the Complexity of Networks: Kolmogorov Complexity versus Entropy, Complexity, Volume 2017 (2017), Article ID 3250301, p. 5.
Simon Plouffe, Copeland-Erdos constant, the primes concatenated.
Simon Plouffe, Copeland-Erdos constant, the primes concatenated.
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant.

Cf. A030168 (continued fraction), A072754 (numerators of convergents), A072755 (denominators of convergents).
Cf. A000040 (primes), A097944 (row lengths if this is read as table), A228355 (digits of the primes listed in reversed order).
Cf. A033307 (Champernowne constant: analog for positive integers instead of primes), A007376 (digits of the integers, considered as infinite word or table), A066716 (decimals of the binary Champernowne constant).
Cf. A165450, A019518, A074721, A073034, A191232, A129808.
Cf. A066747 and A191232: binary Copeland-Erdős constant: decimals and binary digits.
See also A338072.

a033308 n = a033308_list !! (n-1)
a033308_list = concatMap (map (read . return) . show) a000040_list :: [Int]
-- Reinhard Zumkeller, Mar 03 2014

N[Sum[Prime[n]*10^-(n + Sum[Floor[Log[10, Prime[k]]], {k, 1, n}]), {n, 1, 40}], 100] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006 *)
N[Sum[Prime@n*10^-(n + Sum[Floor[Log[10, Prime@k]], {k, n}]), {n, 45}], 106] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006 *)
IntegerDigits //@ Prime@Range@45 // Flatten (* Robert G. Wilson v Oct 03 2006 *)

default(realprecision, 2080); x=0.0; m=-1; forprime (p=2, 4000, n=1+floor(log(p)/log(10)); x=p+x*10^n; m+=n; ); x=x/10^m; for (n=0, 2000, d=floor(x); x=(x-d)*10; write("b033308.txt", n, " ", d)); \\ Harry J. Smith, Apr 30 2009

concat( apply( {row(n)=digits(prime(n))},  [1..99] )) \\ Yields this sequence; row(n) then yields the digits of prime(n) = n-th row of the table, cf. comments. - M. F. Hasler, Oct 25 2019

Equals Sum_{n>=1} prime(n)*10^(-A068670(n)). - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006

Equals Sum_{i>=1} (p_i * 10^(-(Sum_{j=1..i} 1 + floor(log_10(p_j))) )) or Sum_{i>=1} (p_i * 10^(-( i + Sum_{j=1..i} floor(log_10(p_j))) )) or Sum_{i>=1} (p_i * 10^(-( Sum_{j=1..i} ceiling(log_10(1 + p_j))) )). - Daniel Forgues, Mar 26-28 2014

A069090 Primes none of whose proper initial segments are primes.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 41, 43, 47, 61, 67, 83, 89, 97, 101, 103, 107, 109, 127, 149, 151, 157, 163, 167, 181, 401, 409, 421, 443, 449, 457, 461, 463, 467, 487, 491, 499, 601, 607, 631, 641, 643, 647, 653, 659, 661, 683, 691, 809, 811, 821, 823, 827, 829

Offset: 1

Joseph L. Pe, Apr 05 2002

			The proper initial segments of 499 are 4 and 49, none of which are primes. So 499 is a term of the sequence.

Franklin T. Adams-Watters and R. Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from Franklin T. Adams-Watters)
Barry Carter, Table of n, a(n) for n = 1..1411151 (bz2 compressed)
Barry Carter, Mathematica program
StackExchange, Number of digits until a prime is reached

Cf. A074721. [Franklin T. Adams-Watters, Jun 26 2009]

Cf. A000040, A010051, A276707.

import Data.List (inits)
a069090 n = a069090_list !! (n-1)
a069090_list = filter
   (all (== 0) . map (a010051 . read) . init . tail . inits . show)
   a000040_list
-- Reinhard Zumkeller, Mar 11 2014

isA069090 := proc(n)
    local dgs,l ;
    if isprime(n) then
        dgs := convert(n,base,10) ;
        ndgs := nops(dgs) ;
        for l from 1 to ndgs-1 do
            add( op(ndgs+i-l+1,dgs)*10^i,i=0..l-1) ;
            if isprime(%) then
                return false;
            end if;
        end do:
        true ;
    else
        false ;
    end if;
end proc:
for n from 2 to 830 do
    if isA069090(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Dec 15 2016

Select[Prime[Range[200]],NoneTrue[FromDigits/@Table[Take[ IntegerDigits[ #], n],{n,IntegerLength[#]-1}],PrimeQ]&] (* The program uses the NoneTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jul 24 2016 *)

ina(n)=if(!isprime(n),return(0));while(n>9,n\=10;if(isprime(n),return(0)));1 \\ Franklin T. Adams-Watters, Jun 26 2009

from sympy import primerange, isprime
def ok(p):
    s = str(p)
    if len(s) == 1: return True
    return all(not isprime(int(s[:i])) for i in range(1, len(s)))
def aupto(lim):
    alst = []
    for p in primerange(1, lim+1):
        if ok(p): alst.append(p)
    return alst
print(aupto(829)) # Michael S. Branicky, Jul 03 2021

More terms from Franklin T. Adams-Watters, Jun 26 2009

A354839 Beginning with 0, smallest positive integer not yet in the sequence such that the concatenation of two digits of the sequence separated by a comma is prime.

Original entry on oeis.org

0, 2, 3, 1, 7, 9, 70, 5, 30, 20, 21, 10, 22, 31, 11, 12, 32, 33, 13, 14, 15, 34, 16, 17, 18, 35, 36, 19, 71, 37, 38, 39, 72, 90, 23, 73, 74, 75, 91, 76, 77, 92, 93, 78, 94, 79, 700, 24, 100, 25, 95, 96, 101, 97, 98, 99, 701, 102, 300, 26, 103, 104, 105, 301

Offset: 0

Carole Dubois, Jun 08 2022

			a(4)=1 because this is the first number not in the sequence whose first digit is 3 (last digit of a(3)), concatenated with its first digit 1, is prime: 31.
a(14)=31 because this is the first number not in the sequence whose first digit is 2 (last digit of a(13)), concatenated with its first digit 3, is prime: 23.

Cf. A074721, A158652, A152604-A152609, A152136, A152607.

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    aset, k, mink = {0}, 0, 1; yield 0
    for n in count(2):
        k, prevdig = mink, str(k%10)
        while k in aset or not isprime(int(prevdig+str(k)[0])): k += 1
        aset.add(k); yield k
        while mink in aset: mink += 1
print(list(islice(agen(), 64))) # Michael S. Branicky, Jun 09 2022

cp's OEIS Frontend

A151796 Erroneous version of A074721.

Views

Author

Keywords

A033308 Decimal expansion of Copeland-Erdős constant: concatenate primes.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

PARI

Formula

A069090 Primes none of whose proper initial segments are primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Extensions

A354839 Beginning with 0, smallest positive integer not yet in the sequence such that the concatenation of two digits of the sequence separated by a comma is prime.

Views

Author

Keywords

Examples

Crossrefs

Programs

Python