A338072 -id:A338072 - 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

A083758 Lexicographically earliest infinite sequence of distinct primes such that the concatenation of the initial n terms is a prime for all n >= 1.

Original entry on oeis.org

2, 3, 11, 7, 41, 31, 17, 163, 23, 79, 197, 241, 29, 37, 59, 193, 227, 229, 239, 439, 929, 337, 257, 1447, 509, 19, 293, 1723, 1619, 937, 179, 367, 251, 1063, 4241, 1291, 521, 1951, 443, 139, 191, 1753, 1217, 673, 53, 883, 809, 109, 5381, 3733, 311, 967, 449

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

Conjecture: every prime except 5 is a term.

However, after 1000 terms, 13, 47, 61, ... are still missing. A158521 suggests there is no intrinsic reason why 13 should not eventually appear. - N. J. A. Sloane, Oct 21 2020

			2 is a prime.
2||3 = 23 is a prime.
2||3||7 = 3*79 but 2||3||11 = 2311 is a prime
So is 23117.  And so on.

Paul Zimmermann, Table of n, a(n) for n = 1..1479 [First 800 terms from Giorgos Kalogeropoulos; first 1156 terms from Metin Sariyar]

Cf. A051670, A083759, A338072, A338073, A051670.

A083759 Primes arising in A083758.

Original entry on oeis.org

2, 23, 2311, 23117, 2311741, 231174131, 23117413117, 23117413117163, 2311741311716323, 231174131171632379, 231174131171632379197, 231174131171632379197241, 23117413117163237919724129

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 06 2003

Cf. A083758, A338072.

More terms from Sean A. Irvine, Dec 15 2009

A338073 Rearrangement of positive integers except 3 defined by A083758.

Original entry on oeis.org

1, 2, 5, 4, 13, 11, 7, 38, 9, 22, 45, 53, 10, 12, 17, 44, 49, 50, 52, 85, 158, 68, 55, 229, 97, 8, 62, 269, 256, 159, 41, 73, 54, 179, 581, 210, 98, 297, 86, 34, 43, 273, 199, 122, 16, 153, 140, 29, 709, 521, 64, 163, 87, 389, 1145, 106, 164, 231, 26, 177, 28, 464, 51, 258, 398, 391

Offset: 1

N. J. A. Sloane, Oct 20 2020

nonn

This is conjectured to be a permutation of all positive integers except 3.

Apply PrimePi (A000720) to the terms of A083758.

Cf. A000720, A083758, A083759, A338072.

a338073(m)={my(np=1000*m,pused=vectorsmall(np),digp=[]);for(n=1,m,my(found=0);for(k=1,np, if(!pused[k],my(add=digits(prime(k)),pc=concat(digp,add));if(ispseudoprime(fromdigits(pc)),print1(k,", ");digp=pc;pused[k]=1;found=1;break)));if(!found,break))};
a338073(66) \\ Hugo Pfoertner, Oct 21 2020

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

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A083758 Lexicographically earliest infinite sequence of distinct primes such that the concatenation of the initial n terms is a prime for all n >= 1.

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A083759 Primes arising in A083758.

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A338073 Rearrangement of positive integers except 3 defined by A083758.

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