A129808 -id:A129808 - 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

A154513 The prime(n)-th digit of the concatenated composites.

Original entry on oeis.org

6, 8, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 1, 4, 6, 0, 1, 1, 1, 7, 1, 1, 1, 3, 8, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 3, 5, 1, 8, 1, 8, 1, 0, 1, 7, 0, 1, 5, 1, 1, 2, 5, 8, 2, 2, 5, 2, 9, 2, 4, 2, 8, 0, 2, 2, 8, 2, 1, 2, 2, 2, 2, 6, 2, 2, 2, 5, 2, 5, 2, 2, 2, 5

Offset: 1

Juri-Stepan Gerasimov, Jan 11 2009

The concatenated composites are A129808, the initial 1 removed.

Cf. A000040, A002808, A154513, A154520.

a129808 := [1 ] : for n from 4 to 400 do if not isprime(n) then a129808 := [op(a129808), op(ListTools[Reverse](convert(n,base,10))) ] ; fi; od:
for n from 1 do p := ithprime(n) ; printf("%d,", op(p+1,a129808) ) ; od: # R. J. Mathar, Aug 03 2009

nn=700;With[{c=Flatten[IntegerDigits/@Complement[Range[4,nn],Prime[Range[ PrimePi[nn]]]]]}, Flatten[Table[Take[c,{n,n}],{n,Prime[Range[ PrimePi[ nn]]]}]]] (* Harvey P. Dale, Nov 22 2013 *)

a(n) = A129808(prime(n)+1).

Edited and corrected by R. J. Mathar, Aug 03 2009

A154520 The prime(n)-th digit of the concatenated nonprimes.

Original entry on oeis.org

1, 4, 8, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 7, 8, 9, 9, 9, 9, 0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 3, 4, 1, 4, 4, 4, 5, 1, 1, 5, 1, 6, 1, 6, 1, 7, 1, 1, 8, 1, 8, 8, 1, 1, 1, 0, 0, 2, 0, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 3, 3, 2, 4, 2, 4, 4, 5, 2, 2, 2, 6, 7, 7, 2

Offset: 1

Juri-Stepan Gerasimov, Jan 11 2009

The concatenated nonprimes A141468 are A129808 with an initial 0.

Cf. A000040, A141468.

a129808 := [1] : for n from 4 to 400 do if not isprime(n) then a129808 := [op(a129808), op(ListTools[Reverse](convert(n,base,10))) ] ; fi; od:
for n from 1 do p := ithprime(n) ; printf("%d,", op(p-1,a129808) ) ; od: # R. J. Mathar, Aug 03 2009

a(n) = A129808(prime(n)-1) .

Edited and extended by R. J. Mathar, Aug 03 2009

A077305 Partition the concatenation 468910121415161820212224252627283032... of composite numbers into successive strings such that the n-th string is a multiple of prime(n) and > prime(n).

Original entry on oeis.org

4, 6, 8910, 121415, 161820212224, 25262728303233343, 536383940424445464849505152, 5455565, 758606263, 6465666, 8697072747576777880818, 2848586878890919293949, 59698991001021041051061081101111121141151161171181191201

Offset: 1

Amarnath Murthy, Nov 03 2002

			a(4) = 121415 is a multiple of prime(4) = 7.

Cf. A077293, A077294, A077295, A077296, A077297, A077298, A077299, A077300, A077301, A077302, A077303, A077304.

Cf. A129808.

More terms from Franklin T. Adams-Watters, May 30 2006

A129112 Decimal expansion of constant equal to concatenated semiprimes.

Original entry on oeis.org

4, 6, 9, 1, 0, 1, 4, 1, 5, 2, 1, 2, 2, 2, 5, 2, 6, 3, 3, 3, 4, 3, 5, 3, 8, 3, 9, 4, 6, 4, 9, 5, 1, 5, 5, 5, 7, 5, 8, 6, 2, 6, 5, 6, 9, 7, 4, 7, 7, 8, 2, 8, 5, 8, 6, 8, 7, 9, 1, 9, 3, 9, 4, 9, 5, 1, 0, 6, 1, 1, 1, 1, 1, 5, 1, 1, 8, 1, 1, 9, 1, 2, 1, 1, 2, 2, 1, 2, 3, 1, 2, 9

Offset: 1

Jonathan Vos Post, May 24 2007

Is this, as Copeland and Erdos (1946) showed for the Copeland-Erdos constant, a normal number in base 10? I conjecture that it is, despite the fact that the density of odd semiprimes exceeds the density of even semiprimes. What are the first few digits of the continued fraction of this constant? What are the positions of the first occurrence of n in the continued fraction? What are the incrementally largest terms and at what positions do they occur?
Coincides up to n=15 with concatenation of A046368. - M. F. Hasler, Oct 01 2007
Indeed, a theorem of Copeland & Erdős proves that this constant is 10-normal. - Charles R Greathouse IV, Feb 06 2015

			4.691014152122252633343538394649515557586265...

A. H. Copeland and P. Erdős, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), pp. 857-860.
Eric Weisstein's World of Mathematics, Copeland-Erdos Constant.

Cf. A001358, A019518, A030168, A033308 = decimal expansion of Copeland-Erdos constant: concatenate primes, A033309-A033311, A129808.

Flatten[IntegerDigits/@Select[Range[200],PrimeOmega[#]==2&]] (* Harvey P. Dale, Jan 17 2012 *)

print1(4); for(n=6,129, if(bigomega(n)==2, d=digits(n); for(i=1,#d, print1(", "d[i])))) \\ Charles R Greathouse IV, Feb 06 2015

A154521 The prime(n)-th digit of the concatenated positive nonprimes.

Original entry on oeis.org

4, 6, 9, 0, 4, 5, 8, 0, 2, 6, 7, 2, 4, 5, 8, 2, 6, 8, 1, 4, 5, 8, 2, 5, 0, 4, 5, 7, 8, 1, 0, 2, 5, 6, 2, 0, 0, 1, 1, 4, 6, 1, 0, 2, 2, 2, 2, 3, 4, 3, 6, 0, 4, 4, 6, 8, 2, 5, 5, 6, 5, 1, 6, 9, 7, 1, 7, 8, 4, 8, 6, 8, 9, 9, 9, 0, 2, 0, 6, 0, 3, 1, 7, 1, 2, 1, 4, 2, 0, 3, 2, 7, 4, 3, 4, 7, 9, 4, 5, 6, 6, 8, 2, 4, 7

Offset: 1

Juri-Stepan Gerasimov, Jan 11 2009

The digits of A129808 at prime indices.

Cf. A000040, A154513, A154520.

a129808 := [1] : for n from 4 to 400 do if not isprime(n) then a129808 := [op(a129808), op(ListTools[Reverse](convert(n,base,10))) ] ; fi; od:
for n from 1 do p := ithprime(n) ; printf("%d,", op(p,a129808) ) ; od: # R. J. Mathar, Aug 03 2009

a(n) = A129808(prime(n)) .

Edited and corrected by R. J. Mathar, Aug 03 2009

cp's OEIS Frontend

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

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A154513 The prime(n)-th digit of the concatenated composites.

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A154520 The prime(n)-th digit of the concatenated nonprimes.

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A077305 Partition the concatenation 468910121415161820212224252627283032... of composite numbers into successive strings such that the n-th string is a multiple of prime(n) and > prime(n).

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A129112 Decimal expansion of constant equal to concatenated semiprimes.

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A154521 The prime(n)-th digit of the concatenated positive nonprimes.

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