A074785 -id:A074785 - cp's OEIS frontend

A361972 Decimal expansion of lim_{n->oo} ( Sum_{k=2..n} 1/(k*log(k)) - log(log(n)) ).

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

Offset: 0

Bernard Schott, Apr 08 2023

Let u(n) = Sum_{k=2..n} 1/(k*log(k)) - log(log(n)), then (u(n)) is strictly decreasing and lower bounded by -log(log(2)) = A074785, so (u(n)) is convergent, while the series v(n) = Sum_{k=2..n} 1/(k*log(k)) diverges (see Mathematics Stack Exchange link).

Compare with w(n) = Sum_{k=1..n} 1/k - log(n) that converges (A001620), while the harmonic series H(n) = Sum_{k=1..n} 1/k diverges.

			0.79467864545289940220389796...

J. Guégand and M.-A. Maingueneau, Exercices d'Analyse, Exercice 1.18 p. 23, 1988, Classes Préparatoires aux Grandes Ecoles, Ellipses.

Mathematics Stack Exchange, Infinite series Sum_{n=2..oo} 1/(n*log(n)).

Cf. A001620, A074785, A241005.

limit(sum(1/(k*log(k)), k=2..n) - log(log(n)), n = infinity);

Limit_{n->oo} 1/(2*log(2)) + 1/(3*log(3)) + ... + 1/(n*log(n)) - log(log(n)).

Equals A241005 - log(log(2)) = A241005 + A074785. - Amiram Eldar, Apr 08 2023

A194562 Decimal expansion of log(log(3)).

Original entry on oeis.org

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

Offset: 0

Kausthub Gudipati, Sep 20 2011

			0.09404782761669901617433433208449399278533802961841...

Cf. A002391, A074785.

evalf(log(log(3.0)) ; # R. J. Mathar, Sep 20 2011

RealDigits[Log[Log[3]], 10, 120, -1][[1]] (* Amiram Eldar, Jun 15 2023 *)

log(log(3)) \\ Charles R Greathouse IV, Oct 21 2021

Equals log(A002391).

A059200 Engel expansion of -log(log(2)) = 0.36651292... .

Original entry on oeis.org

3, 11, 11, 23, 62, 66, 466, 1450, 7617, 95677, 100963, 153329, 966054, 4744661, 23899231, 25086529, 52363821, 100389201, 201892089, 261170111, 312778184, 527002514, 1235004065, 1623652949, 2309078745, 8274570969

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmänner in Marburg, 1913, pp. 190-191.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A074785.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[-Log[Log[2]], 7!], 100] (* Modified by G. C. Greubel, Dec 28 2016 *)

A059567 Beatty sequence for 1 - log(log(2)).

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 30, 31, 32, 34, 35, 36, 38, 39, 40, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 86, 87, 88, 90, 91, 92, 94, 95, 97

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence
Index entries for sequences related to Beatty sequences

Beatty complement is A059568.

Cf. A074785.

Floor[Range[100]*(1 - Log[Log[2]])] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=1 - log(log(2)); for (n = 1, 2000, write("b059567.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*(1 - log(log(2)))). - Michel Marcus, Jan 04 2015

A059568 Beatty sequence for 1 - 1/log(log(2)).

Original entry on oeis.org

3, 7, 11, 14, 18, 22, 26, 29, 33, 37, 41, 44, 48, 52, 55, 59, 63, 67, 70, 74, 78, 82, 85, 89, 93, 96, 100, 104, 108, 111, 115, 119, 123, 126, 130, 134, 137, 141, 145, 149, 152, 156, 160, 164, 167, 171, 175, 178, 182, 186, 190, 193, 197, 201, 205, 208, 212, 216

Offset: 1

Mitch Harris, Jan 22 2001

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, and Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Beatty complement is A059567.

Cf. A074785.

Floor[Range[100]*(1 - 1/Log[Log[2]])] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=1 - 1/log(log(2)); for (n = 1, 2000, write("b059568.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*(1 - 1/log(log(2)))). - Michel Marcus, Jan 04 2015

A248472 Decimal expansion of C_1 = gamma + log(log(2)) - 2*Ei(-log(2)), one of the Tauberian constants, where Ei is the exponential integral function.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 27 2014

			0.96804483044204448704848730112285492269036397005924632964...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Steven R. Finch, Tauberian Constants, August 30, 2004 [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 68.
Eric Weisstein's MathWorld, Exponential Integral

Cf. A001620, A074785, A249385.

evalf(gamma + log(log(2)) - 2*Ei(-log(2)), 120); # Vaclav Kotesovec, Oct 27 2014

C1 = EulerGamma + Log[Log[2]] - 2*ExpIntegralEi[-Log[2]]; RealDigits[C1, 10, 103] // First

Euler + log(log(2)) + 2*eint1(log(2)) \\ Altug Alkan, Sep 05 2018

C_1 also equals gamma + log(log(2)) + 2*Gamma(0, log(2)), where Gamma is the incomplete gamma function.

A171990 Least integer a(n) for which the iterated function log, iterated n times, is defined.

Original entry on oeis.org

1, 2, 3, 16, 3814280

Offset: 1

Alexis Monnerot-Dumaine, Jan 21 2010

nonn

Log(a(1)) is defined if a(1) > 0, so a(1) = 1.
Log(log(a(2))) is defined if log(a(2)) > 0 => a(2) > 1 => a(2) = 2.
The sequence grows rapidly: a(6) = 2.33150...10^1656520, and is too large to include here.

			a(2) = 2 because log(log(2)) is defined and log(log(1)) is not;
a(3) = 3 because log(log(log(3))) is defined;
a(4) = 16 because log(log(log(log(16)))) is defined.
From _Robert G. Wilson v_, Jul 05 2022: (Start)
a(3) = ceiling(A001113).
a(4) = ceiling(A073226).
a(5) = ceiling(A073227).
a(6) = ceiling(A085667). (End)

Cf. A074785 (log(log(2))), A085667, A004002, A001113, A073226, A073227, A085667.

a(n) = my(k=1); while(1, my(s=k, i=0); while(s > 0, s=log(s); if(s > 0, i++)); if(i==n-1, return(k)); k++) \\ Felix Fröhlich, Nov 22 2015

For n > 2, a(n) = ceiling(e^(e^(...))) where e appears n-2 times.

A193019 -log(log(2)) / log(2).

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Jul 14 2011

This is the value for which 2^x - x is minimized.

			-log(log(2)) / log(2) = .5287663729448976142474977797788....

Cf. A002162, A074785.

A196565 Decimal expansion of log(log(4)).

Original entry on oeis.org

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

Offset: 0

Kausthub Gudipati, Oct 04 2011

			0.3266342599782809824047929632255070986212366865231499910670022958228...

Cf. A002162, A016627, A074785, A194562, A074785.

RealDigits[Log[Log[4]],10,120][[1]] (* Harvey P. Dale, Aug 01 2021 *)

log(log(4)) \\ Charles R Greathouse IV, May 15 2019

From Amiram Eldar, Jun 12 2023: (Start)
Equals log(A016627).
Equals log(2) + log(log(2)) = A002162 - A074785. (End)

A196566 Decimal expansion of log(log(5)).

Original entry on oeis.org

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

Offset: 0

Kausthub Gudipati, Oct 04 2011

			log(log(5)) = 0.4758849953271106210...

Cf. A016628, A194562, A074785.

RealDigits[Log[Log[5]],10,120][[1]] (* Harvey P. Dale, Mar 19 2019 *)

log(log(5)) \\ Charles R Greathouse IV, May 15 2019

cp's OEIS Frontend

A361972 Decimal expansion of lim_{n->oo} ( Sum_{k=2..n} 1/(k*log(k)) - log(log(n)) ).

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A194562 Decimal expansion of log(log(3)).

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A059200 Engel expansion of -log(log(2)) = 0.36651292... .

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A059567 Beatty sequence for 1 - log(log(2)).

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A059568 Beatty sequence for 1 - 1/log(log(2)).

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A248472 Decimal expansion of C_1 = gamma + log(log(2)) - 2*Ei(-log(2)), one of the Tauberian constants, where Ei is the exponential integral function.

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A171990 Least integer a(n) for which the iterated function log, iterated n times, is defined.

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A193019 -log(log(2)) / log(2).

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