A072754 -id:A072754 - 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

A030168 Continued fraction for Copeland-Erdős constant 0.235711... (concatenate primes).

Original entry on oeis.org

0, 4, 4, 8, 16, 18, 5, 1, 1, 1, 1, 7, 1, 1, 6, 2, 9, 58, 1, 3, 4, 2, 2, 1, 1, 2, 1, 4, 39, 4, 4, 5, 2, 1, 1, 87, 16, 1, 2, 1, 2, 1, 1, 3, 1, 8, 1, 3, 1, 1, 6, 1, 13, 27, 1, 1, 3, 1, 41, 1, 2, 1, 1, 19, 1, 1, 1, 1, 3, 1, 1, 484, 1, 4, 1, 19, 3, 6, 8, 1, 5, 1, 17, 9, 2, 3, 5, 25, 1468, 1, 1, 3, 1

Offset: 0

Eric W. Weisstein

			0.23571113171923293137414347... = 0 + 1/(4 + 1/(4 + 1/(8 + 1/(16 + ...))))

M. F. Hasler, Table of n, a(n) for n = 0..5000
G. Xiao, Contfrac
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant
Index entries for continued fractions for constants

Cf. A033308 (decimal expansion), A072754 (numerators of convergents), A072755 (denominators of convergents).

Take[ ContinuedFraction@ FromDigits[{Flatten[ IntegerDigits[ Prime@Range@ 47]], 0}], 95] (* Robert G. Wilson v, Oct 17 2013 *)

s=concat(vector(2000,i,Str(prime(i)))); c=contfrac(eval(s)/10^#s);
c2=contfrac((eval(s)+10^9)/10^#s);
for(i=1,#c, c[i]!=c2[i] & return(Str("Terms may be wrong for n>="i-1));
write("b030168.txt",i-1," ",c[i])) \\ M. F. Hasler, Oct 13 2009

{ default(realprecision, 2100); x=0.0; m=0; forprime (p=2, 4000, n=1+floor(log(p)/log(10)); x=p+x*10^n; m+=n; ); x=contfrac(x/10^m); for (n=1, 2001, write("b030168.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 30 2009

A088317 a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.

Original entry on oeis.org

1, 4, 33, 268, 2177, 17684, 143649, 1166876, 9478657, 76996132, 625447713, 5080577836, 41270070401, 335241141044, 2723199198753, 22120834731068, 179689877047297, 1459639851109444, 11856808685922849, 96314109338492236, 782369683393860737, 6355271576489378132, 51624542295308885793

Offset: 0

Nikolay V. Kosinov (kosinov(AT)unitron.com.ua), Nov 06 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Tanya Khovanova, Recursive Sequences.
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,1).

Cf. A002018, A002190, A013192, A028576, A041024, A041027.

Cf. A053120, A058153, A058155, A072754, A075132.

[n le 2 select 4^(n-1) else 8*Self(n-1) +Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 13 2022

LinearRecurrence[{8,1},{1,4},30] (* or *) With[{c=Sqrt[17]},Simplify/@ Table[1/2 (c-4)((c+4)^n-(4-c)^n (33+8c)),{n,30}]] (* Harvey P. Dale, May 07 2012 *)

a[0]:1$ a[1]:4$ a[n]:=8*a[n-1]+a[n-2]$ A088317(n):=a[n]$
makelist(A088317(n),n,0,20); /* Martin Ettl, Nov 12 2012 */

A088317=BinaryRecurrenceSequence(8,1,1,4)
[A088317(n) for n in range(31)] # G. C. Greubel, Dec 13 2022

a(n) = ( (4+sqrt(17))^n + (4-sqrt(17))^n )/2.
a(n) = A086594(n)/2.
Lim_{n -> oo} a(n+1)/a(n) = 4 + sqrt(17).
From Paul Barry, Nov 15 2003: (Start)
E.g.f.: exp(4*x)*cosh(sqrt(17)*x).
a(n) = Sum_{k=0..floor(n/2)} C(n, 2*k)*17^k*4^(n-2*k).
a(n) = (-i)^n * T(n, 4*i) with T(n, x) Chebyshev's polynomials of the first kind (see A053120) and i^2=-1. (End)
a(n) = A041024(n-1), n>0. - R. J. Mathar, Sep 11 2008
G.f.: (1-4*x)/(1-8*x-x^2). - Philippe Deléham, Nov 16 2008 and Nov 20 2008
a(n) = (1/2)*((33+8*sqrt(17))*(4-sqrt(17))^(n+2) + (33-8*sqrt(17))*(4+sqrt(17))^(n+2)). - Harvey P. Dale, May 07 2012

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

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A030168 Continued fraction for Copeland-Erdős constant 0.235711... (concatenate primes).

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A088317 a(n) = 8*a(n-1) + a(n-2), starting with a(0) = 1 and a(1) = 4.

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A072755 Denominators of continued fraction convergents to Copeland-Erdos constant.

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