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%I #76 Sep 26 2023 06:22:59
%S 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,
%T 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,
%U 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
%N Decimal expansion of Copeland-Erdős constant: concatenate primes.
%C 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
%C 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
%C Named after the American mathematician Arthur Herbert Copeland (1898-1970) and the Hungarian mathematician Paul Erdős (1913-1996). - _Amiram Eldar_, May 29 2021
%D 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.
%D Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
%H T. D. Noe, <a href="/A033308/b033308.txt">Table of n, a(n) for n = 0..2000</a>
%H John M. Campbell, <a href="https://arxiv.org/abs/2309.13520">The prime-counting Copeland-Erdős constant</a>, arXiv:2309.13520 [math.NT], 2023.
%H A. H. Copeland and P. Erdős, <a href="http://dx.doi.org/10.1090/S0002-9904-1946-08657-7">Note on Normal Numbers</a>, Bull. Amer. Math. Soc., Vol. 52, No. 10 (1946), pp. 857-860.
%H Mikołaj Morzy, Tomasz Kajdanowicz, and Przemysław Kazienko, <a href="https://dx.doi.org/10.1155/2017/3250301">On Measuring the Complexity of Networks: Kolmogorov Complexity versus Entropy</a>, Complexity, Volume 2017 (2017), Article ID 3250301, p. 5.
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap13.html">Copeland-Erdos constant, the primes concatenated</a>.
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/copelan.txt">Copeland-Erdos constant, the primes concatenated</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Copeland-ErdosConstant.html">Copeland-Erdős Constant</a>.
%F Equals Sum_{n>=1} prime(n)*10^(-A068670(n)). - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006
%F 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
%e 0.235711131719232931374143475359616771737983899710110310710911312...
%t 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 *)
%t 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 *)
%t IntegerDigits //@ Prime@Range@45 // Flatten (* _Robert G. Wilson v_ Oct 03 2006 *)
%o (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
%o (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
%o (Haskell)
%o a033308 n = a033308_list !! (n-1)
%o a033308_list = concatMap (map (read . return) . show) a000040_list :: [Int]
%o -- _Reinhard Zumkeller_, Mar 03 2014
%Y Cf. A030168 (continued fraction), A072754 (numerators of convergents), A072755 (denominators of convergents).
%Y Cf. A000040 (primes), A097944 (row lengths if this is read as table), A228355 (digits of the primes listed in reversed order).
%Y 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).
%Y Cf. A165450, A019518, A074721, A073034, A191232, A129808.
%Y Cf. A066747 and A191232: binary Copeland-Erdős constant: decimals and binary digits.
%Y See also A338072.
%K nonn,cons,base
%O 0,1
%A _Eric W. Weisstein_
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