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

0,1

Comments

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

References

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.

Links

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.

Formula

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

Example

0.235711131719232931374143475359616771737983899710110310710911312...

Mathematica

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 *)

Prog

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

Crossrefs

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.

Sequence in context: A113493 A060420 A077648 * A134690 A295868 A228355

Adjacent sequences: A033305 A033306 A033307 * A033309 A033310 A033311

Keyword

nonn,cons,base

Author

Eric W. Weisstein

Status

approved