A097944 -id:A097944 - 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

A068670 Number of digits in the concatenation of first n primes.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154

Offset: 0

Eugene McDonnell (eemcd(AT)mac.com), Jan 18 2004

Partial sums of A097944. - Lekraj Beedassy, Aug 26 2008

			a(5) is 6 because concatenating the first five primes gives 235711, which has six digits.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000. (Initial 0 added by _M. F. Hasler_, Nov 02 2019.)
Eric Weisstein's World of Mathematics, Copeland-Erdos Constant

Cf. A019518, A097944 (number of decimal digits of the primes).

Cf. A033308 (decimal expansion of the Copeland-Erdos constant).

a068670:=func< n | n + &+[ Floor(Log(10, NthPrime(k))): k in [1..n] ] >; [ a068670(n): n in [1..70] ];

Table[n + Sum[Floor[Log[10, Prime[k]]], {k, 1, n}], {n, 1, 90}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006 *)
Accumulate[IntegerLength[Prime[Range[70]]]] (* Harvey P. Dale, Jul 01 2012 *)

A68670=List(0); A068670(n)={for(N=#A68670,n, listput(A68670, A68670[N] + A097944(N))); A68670[n+1]} \\ M. F. Hasler, Oct 24 2019

from sympy import sieve
from itertools import accumulate, chain
def f(, n): return  + len(str(n))
def agen(): yield from accumulate(chain((0,), (p for p in sieve)), f)
print(list(islice(agen(), 62))) # Michael S. Branicky, Feb 03 2023

a(n) = Sum_{i=1..n} ceiling(log_10(1 + prime(i))). - Daniel Forgues, Apr 02 2014

Extended to a(0) = 0 by M. F. Hasler, Oct 24 2019

A074462 Average digit (rounded up) in the decimal expansion of prime(n).

Original entry on oeis.org

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

Offset: 1

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 23 2002

			The prime numbers begin with 2,3,5,7,11,13,17,19,23,... so the average digits rounded up are 2, 3, 5, 7, (1+1)/2=1, (1+3)/2=2, (1+7)/2=4, (1+9)/2=5, ceiling((2+3)/2)=3, ...

Cf. A000040, A004427, A074461, A073342.

Cf. A007605, A097944.

Table[Ceiling[Mean[IntegerDigits[p]]],{p,Prime[Range[120]]}] (* Harvey P. Dale, Nov 06 2022 *)

a(n) = my(d=digits(prime(n))); ceil(vecsum(d)/#d); \\ Michel Marcus, Apr 23 2022

a(n) = ceiling(A007605(n)/A097944(n)). - R. J. Mathar, Sep 23 2008

a(n) = A004427(A000040(n)). - Reinhard Zumkeller, May 27 2010

Offset changed to 1, cf. to A073342 added, and extended by R. J. Mathar, Sep 23 2008

A228355 Write the primes backwards in base 10 and juxtapose their digits.

Original entry on oeis.org

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

Offset: 0

Vincenzo Librandi, Aug 21 2013

Also, decimal expansion of the constant 0.235711317191329213731434743595...

The same sequence, but with different indexing, would arise for the table whose n-th row lists the A097944(n) digits of the n-th prime A000040(n) from right to left. - M. F. Hasler, Oct 24 2019

Vincenzo Librandi, Table of n, a(n) for n = 0..2000

Cf. A004087, A007376, A033308.

Reverse[IntegerDigits//@ Reverse@Prime@Range@50//Flatten]

fromdigits( A=concat(apply( row(n)=Vecrev(digits(prime(n))), [1..default(realprecision)])))*.1^#A \\ Result = the constant, A = sequence of digits, row(n) = n-th row of the table. - M. F. Hasler, Oct 24 2019

A060417 Number of different decimal digits in n-th prime.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Apr 05 2001

			The maximum is 10. Thus 1234567891 is a prime with 9 and 123456789011 with 10 different decimal digits.

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

Cf. A055642, A097944.

import Data.List (nub, sort)
a060417 = length . nub . show . a000040
-- Reinhard Zumkeller, Apr 08 2012

Table[Length[Union[IntegerDigits[Prime[w]]]], {w, 1, 1000}]

A073342 Average digit (rounded to the nearest integer) in the decimal expansion of n-th prime.

Original entry on oeis.org

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

Offset: 1

Y. Kelly Itakura (yitkr(AT)mta.ca), Aug 23 2002

			For the prime 107 we have nearest((1+0+7)/3)=nearest(8/3)=3.

Cf. A000040, A074462.

Floor[Mean[IntegerDigits[#]]+1/2]&/@Prime[Range[120]] (* Harvey P. Dale, Nov 22 2011 *)

a(n)=round(A007605(n)/A097944(n)). - R. J. Mathar, Sep 23 2008

Changed offset to 1, added Cf. to A074462 and extended. - R. J. Mathar, Sep 23 2008

A175762 Primes with an arithmetic mean of digits which is also prime.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 31, 37, 59, 73, 1061, 1151, 1223, 1289, 1487, 1511, 1559, 1601, 1667, 1847, 1973, 1999, 2099, 2141, 2213, 2297, 2411, 2459, 2477, 2549, 2657, 2693, 2729, 2819, 2837, 2909, 2927, 2963, 3023, 3041, 3089, 3203, 3221, 3359, 3449, 3467, 3539, 3557

Offset: 1

Juri-Stepan Gerasimov, Aug 29 2010

			a(10)=73 because (7+3)/2=5=prime. a(12)=1151 because (1+1+5+1)/4=2=prime.

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

Cf. A000040, A007605, A061383.

Select[Prime[Range[500]],PrimeQ[Mean[IntegerDigits[#]]]&] (* Harvey P. Dale, Jan 16 2013 *)

{A000040(i): A007605(i)/A097944(i) in A000040}.

Corrected (59 inserted, 1061 inserted etc.) by R. J. Mathar, Sep 24 2010

A278959 Length of the string that is generated by the concatenation of all the prime numbers < n (where n >= 0).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 3, 3, 4, 4, 4, 4, 6, 6, 8, 8, 8, 8, 10, 10, 12, 12, 12, 12, 14, 14, 14, 14, 14, 14, 16, 16, 18, 18, 18, 18, 18, 18, 20, 20, 20, 20, 22, 22, 24, 24, 24, 24, 26, 26, 26, 26, 26, 26, 28, 28, 28, 28, 28, 28, 30

Offset: 0

Indranil Ghosh, Dec 02 2016

In the following Python program, the algorithm based on the sieve of Eratosthenes is used to generate the primes.

			For n=15, the primes < n are 2,3,5,7,11,13. So the concatenated string is "23571113", which has length=8. a(n)=8.

Indranil Ghosh, Table of n, a(n) for n = 0..100000
Wikipedia, Sieve Of Eratosthenes

Cf. A000040, A068670, A097944.

Join[{0},Accumulate[Table[If[PrimeQ[n],IntegerLength[n],0],{n,0,60}]]] (* Harvey P. Dale, Mar 04 2023 *)

def p(n):
    if n<=2:
        return 0
    s=1
    l = [True] * n
    for i in range(3,int(n**0.5)+1,2):
        if l[i]:
            l[i*i::2*i]=[False]*((n-i*i-1)//(2*i)+1)
    for i in range(3,n,2):
            if l[i]:
                s+=len(str(i))
    return s
for i in range(0, 100001):
    print(f'{i} {p(i)}')

cp's OEIS Frontend

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

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A068670 Number of digits in the concatenation of first n primes.

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A074462 Average digit (rounded up) in the decimal expansion of prime(n).

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A228355 Write the primes backwards in base 10 and juxtapose their digits.

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A060417 Number of different decimal digits in n-th prime.

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A073342 Average digit (rounded to the nearest integer) in the decimal expansion of n-th prime.

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Mathematica

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A175762 Primes with an arithmetic mean of digits which is also prime.

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