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A033308
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Decimal expansion of Copeland-Erdős constant: concatenate primes.
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41
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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
(list;
constant;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,1
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COMMENTS
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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
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REFERENCES
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G. Harman, One hundred years of normal numbers, in M. A. Bennett et al., eds., Number Theory for the Millennium, II (Urbana, IL, 2000), 149-166, A K Peters, Natick, MA, 2002.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..2000
A. H. Copeland and P. Erdős, Note on Normal Numbers, Bull. Amer. Math. Soc. 52, 857-860, 1946.
Simon Plouffe, Copeland-Erdos constant, the primes concatenated
Simon Plouffe, Copeland-Erdos constant, the primes concatenated
Eric Weisstein's World of Mathematics, Copeland-Erdos Constant
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FORMULA
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Equals sum(n=1..inf, prime(n)*10^-A068670(n)). - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 12 2006
Equals sum(i=1..inf, p_i * 10^-( sum(j=1..i, 1 + floor(log_10(p_j))) )) or sum(i=1..inf, p_i * 10^-( i + sum(j=1..i, floor(log_10(p_j))) )) or sum(i=1..inf, p_i * 10^-( sum(j=1..i, ceiling(log_10(1 + p_j))) )). - Daniel Forgues, Mar 26-28 2014
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EXAMPLE
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0.235711131719232931374143475359616771737983899710110310710911312...
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MATHEMATICA
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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 *)
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PROG
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(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
(Haskell)
a033308 n = a033308_list !! (n-1)
a033308_list = concatMap (map (read . return) . show) a000040_list :: [Int]
-- Reinhard Zumkeller, Mar 03 2014
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CROSSREFS
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Cf. A030168 (continued fraction), A072754 (numerators of convergents), A072755 (denominators of convergents).
Cf. A033307, A000040, A165450, A019518, A074721, A073034.
Sequence in context: A113493 A060420 A077648 * A134690 A228355 A065859
Adjacent sequences: A033305 A033306 A033307 * A033309 A033310 A033311
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KEYWORD
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nonn,cons,base
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AUTHOR
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Eric W. Weisstein
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STATUS
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approved
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