A030168 -id:A030168 - cp's OEIS frontend

A059102 Number of terms of the fractional part of A030168 for which the geometric mean produces increasingly better approximations to Khinchin's constant.

1, 9, 10, 12, 13, 26, 48, 49, 51, 65, 66, 67, 1330, 1331, 1332, 1336, 1337, 1338, 1339, 1340, 1431, 5666, 10331, 537086, 537095, 538045, 576768, 576789

Offset: 1

Hans Havermann, Feb 13 2001

nonn

Next term > 3307254.

			The geometric mean of 67 terms (Khinchin + 0.0128639) is not bettered until we calculate the geometric mean of 1330 terms (Khinchin + 0.011153).

Cf. A030168, A048613.

p = Drop[{A030168}, 1]; q = N[1, 100]; r = p[[1]] + 1; t = {}; Do[q = q*p[[i]]; g = q^(1/i) - Khinchin; If[Abs[g] < r, r = Abs[g]; t = Append[t, i]], {i, 1, Length[p]}]; t

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

Original entry on oeis.org

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

A033309 Position of first occurrence of n in continued fraction for Copeland-Erdős constant.

Original entry on oeis.org

8, 16, 20, 2, 7, 15, 12, 4, 17, 254, 98, 154, 53, 212, 253, 5, 83, 6, 64, 353, 1049, 555, 562, 690, 88, 334, 54, 1795, 621, 392, 1101, 1203, 715, 3929, 628, 138, 257, 648, 29, 329, 59, 2417, 444, 1353, 342, 1348, 911, 3043, 3870, 840, 990, 681, 229, 689, 749, 2311, 8254, 18

Offset: 1

Eric W. Weisstein

nonn

This version uses an incorrect c.f. term indexing of [a_1; a_2, ...] instead of [a_0; a_1, ...]; see A224891 for correctly indexed version.

Klaus Brockhaus, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant Continued Fraction

Cf. A033308, A030168.

Cf. A224891 (= a(n) - 1).

a(n) = A224891(n) + 1.

More terms from Carl Najafi, May 07 2011

A072754 Numerators of continued fraction convergents to Copeland-Erdos constant.

Original entry on oeis.org

0, 1, 4, 33, 532, 9609, 48577, 58186, 106763, 164949, 271712, 2066933, 2338645, 4405578, 28772113, 61949804, 586320349, 34068530046, 34654850395, 138033081231, 586787175319, 1311607431869, 3210002039057, 4521609470926

Offset: 0

Rick L. Shepherd, Jul 09 2002

Cf. A072755 (denominators), A030168 (continued fraction), A033308 (decimal expansion).

for(n=1,30, v=vector(n,i, A030168[i]); print1(contfracpnqn(v)[1,1],","))

A033310 Incrementally largest terms in continued fraction for Copeland-Erdős constant.

Original entry on oeis.org

0, 4, 8, 16, 18, 58, 87, 484, 1468, 3955, 6986, 17474, 26174, 37084, 73646, 89141, 187645, 542374, 551071, 644045, 925191, 2037817, 4576230, 13320898, 16917960, 190435991, 230132891, 407588556

Offset: 1

Eric W. Weisstein

Eric Weisstein's World of Mathematics, Copeland-Erdős Constant Continued Fraction

Cf. A033308, A033309, A030168.
Cf. A224890 (positions of incrementally largest terms).
Cf. A030168 (continued fraction for Copeland-Erdős constant).

Extended by Eric W. Weisstein, Mar 25 2009

a(25)-a(28) from Eric W. Weisstein, Jul 24 2013

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

Original entry on oeis.org

1, 4, 17, 140, 2257, 40766, 206087, 246853, 452940, 699793, 1152733, 8768924, 9921657, 18690581, 122065143, 262820867, 2487452946, 144535091735, 147022544681, 585602725778, 2489433447793, 5564469621364, 13618372690521

Offset: 0

Rick L. Shepherd, Jul 09 2002

Cf. A072754 (numerators), A030168 (continued fraction), A033308 (decimal expansion).

for(n=1,30, v=vector(n,i, A030168[i]); print1(contfracpnqn(v)[2,1],","))

A033311 Positions of incrementally largest terms in continued fraction for Copeland-Erdős constant.

Original entry on oeis.org

1, 2, 4, 5, 6, 18, 36, 72, 89, 1557, 3728, 4730, 27642, 60322, 90720, 104378, 107183, 241453, 453796, 679114, 901262, 934243, 1298092, 5996127, 7360332, 76543031, 299039550, 382621530

Offset: 1

Eric W. Weisstein

This version uses an incorrect c.f. term indexing of [a_1; a_2, ...] instead of [a_0; a_1, ...]; see A224890 for correctly indexed version.

Eric Weisstein's World of Mathematics, Copeland-Erdős Constant Continued Fraction

Cf. A033308, A033309, A033310, A030168.

Cf. A224890 (= a(n) - 1).

a(n) = A224890(n) + 1.

More terms from Eric W. Weisstein, Mar 25 2009

A224890 Positions of incrementally largest terms in continued fraction for Copeland-Erdős constant.

Original entry on oeis.org

0, 1, 3, 4, 5, 17, 35, 71, 88, 1556, 3727, 4729, 27641, 60321, 90719, 104377, 107182, 241452, 453795, 679113, 901261, 934242, 1298091, 5996126, 7360331, 76543030, 299039549, 382621529

Offset: 1

Eric W. Weisstein, Jul 24 2013

Correctly indexed version of A033311.

			The c.f. of the Copeland-Erdős constant is [a_0; a_1, a_2, ...] = [0; 4, 4, 8, 16, 18, 5, 1, ...], so record terms occur at positions 0, 1, 3, 4, 5, ...

Eric Weisstein's World of Mathematics, Copeland-Erdős Constant Continued Fraction

Cf. A033311 (= a(n) + 1).
Cf. A033310 (incrementally largest terms).
Cf. A030168 (continued fraction for Copeland-Erdős constant).

a(n) = A033311(n) - 1.

a(25)-a(28) from Eric W. Weisstein, Jul 24 2013

A224891 Position of first occurrence of n in continued fraction for Copeland-Erdos constant.

Original entry on oeis.org

7, 15, 19, 1, 6, 14, 11, 3, 16, 253, 97, 153, 52, 211, 252, 4, 82, 5, 63, 352, 1048, 554, 561, 689, 87, 333, 53, 1794, 620, 391, 1100, 1202, 714, 3928, 627, 137, 256, 647, 28, 328, 58, 2416, 443, 1352, 341, 1347, 910, 3042, 3869, 839, 989, 680, 228, 688, 748, 2310, 8253, 17

Offset: 1

Eric W. Weisstein, Jul 24 2013

nonn

Correctly indexed version of A033309.

Smallest numbers not occurring in the first 1,011,597,392 terms of the c.f. are 14731, 15456, 15579, 15869, ...

Eric W. Weisstein, Table of n, a(n) for n = 1..14730
Eric Weisstein's World of Mathematics, Copeland-Erdos Constant Continued Fraction

Cf. A033309 (= a(n) + 1).

Cf. A030168 (continued fraction of the Copeland-Erdos constant).

a(n) = A033309(n) - 1.

A066707 Incrementally largest terms in the continued fraction for the constant given by Sum_{k>=0} A033308(k) / 2^k = 2.89104866587305422....

Original entry on oeis.org

2, 8, 10, 32, 39, 5903, 135598

Offset: 1

Robert G. Wilson v, Jan 14 2002

Previous name: Incrementally largest terms in the continued fraction for Copeland-Erdos constant 0.235711... (concatenate primes).

Appears to be a mistaken version of A033310 in which base 2 has been used in the "concatenation" of the digits in A033308 while still interpreting the digits of A033308 as decimal. - Sean A. Irvine, Nov 03 2023

Cf. A030168, A033310.

a = {}; Do[a = Append[a, IntegerDigits[ Prime[n]]], {n, 1, 5*10^4} ]; b = ContinuedFraction[ N[ FromDigits[{Flatten[a], 0}, 2], 5*10^4]]; c = -1; d = {}; Do[ If[b[[n]] > c, c = b[[n]]; d = Append[d, c]], {n, 1, 48336} ]; d

Name changed to match data by Sean A. Irvine, Nov 03 2023

cp's OEIS Frontend

A059102 Number of terms of the fractional part of A030168 for which the geometric mean produces increasingly better approximations to Khinchin's constant.

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

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A033309 Position of first occurrence of n in continued fraction for Copeland-Erdős constant.

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A072754 Numerators of continued fraction convergents to Copeland-Erdos constant.

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A033310 Incrementally largest terms in continued fraction for Copeland-Erdős constant.

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

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PARI

A033311 Positions of incrementally largest terms in continued fraction for Copeland-Erdős constant.

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A224890 Positions of incrementally largest terms in continued fraction for Copeland-Erdős constant.

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A224891 Position of first occurrence of n in continued fraction for Copeland-Erdos constant.

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