A002162 -id:A002162 - cp's OEIS frontend

A016578 Decimal expansion of log(3/2).

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

Offset: 0

N. J. A. Sloane

			0.4054651081081643819780131154643491365719904234624941976140143...

L. B. W. Jolley, Summation of Series, Dover (1961), eq (102), page 20.

Harry J. Smith, Table of n, a(n) for n = 0..20000
Index entries for transcendental numbers

Cf. A016529, A002391, A002162, A083679.

RealDigits[Log[3/2],10,111][[1]] (* Robert G. Wilson v, Aug 08 2011 *)

default(realprecision, 20080); x=10*log(3/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b016578.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009

Equals Sum {k>=1} 1/(k*3^k). - Robert G. Wilson v, Aug 08 2011
Equals 1/2 - 1/(2*2^2) + 1/(3*2^3) - 1/(4*2^4) + ... [Jolley].
Equals A002391-A002162. - Michel Marcus, Sep 17 2016
From Amiram Eldar, Aug 07 2020: (Start)
Equals 2 * arctanh(1/5).
Equals Integral_{x=0..oo} 1/(2*exp(x) + 1) dx. (End)
log(3/2) = 2*Sum_{n >= 1} 1/(n*P(n, 5)*P(n-1, 5)), where P(n, x) denotes the n-th Legendre polynomial. The first 10 terms of the series gives the approximation log(3/2) = 0.40546510810816438197(04...), correct to 20 decimal places. - Peter Bala, Mar 16 2024
Equals Sum_{n >= 1} (-1)^(n+1) * 5/(n*binomial(2*n, n)*6^n). The n-th term of the series is O(5*sqrt(Pi/n)*1/24^n). - Peter Bala, Mar 04 2025
Equals Integral_{x=0..1} (sqrt(x) - 1)/log(x) dx. - Kritsada Moomuang, Jun 14 2025

A016730 Continued fraction for log(2).

Original entry on oeis.org

0, 1, 2, 3, 1, 6, 3, 1, 1, 2, 1, 1, 1, 1, 3, 10, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 3, 1, 13, 7, 4, 1, 1, 1, 7, 2, 4, 1, 1, 2, 5, 14, 1, 10, 1, 4, 2, 18, 3, 1, 4, 1, 6, 2, 7, 3, 3, 1, 13, 3, 1, 4, 4, 1, 3, 1, 1, 1, 1, 2, 17, 3, 1, 2, 32, 1, 1, 1

Offset: 0

N. J. A. Sloane

Continued fraction for 1/log(2) is the same but without the initial zero.

			log(2) = 0.6931471805599453094... = 0 + 1/(1 + 1/(2 + 1/(3 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 21 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
G. Xiao, Contfrac
Eric Weisstein's World of Mathematics, Natural Logarithm of 2
Index entries for continued fractions for constants

Cf. A120754, A120755, A002162 (decimal expansion).

ContinuedFraction(Log(2)); // G. C. Greubel, Sep 15 2018

ContinuedFraction[Log[2], 80] (* Alonso del Arte, Oct 03 2017 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(2)); for (n=1, 20000, write("b016730.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 21 2009

Offset changed by Andrew Howroyd, Jul 10 2024

A094642 Decimal expansion of log(Pi/2).

Original entry on oeis.org

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

Offset: 0

Jonathan Sondow and Robert G. Wilson v, May 18 2004

			log(Pi/2) = 0.45158270528945486472619522989488214357179467855505...

George Boros and Victor Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, Cambridge, 2004, Chap. 7.
Jonathan Borwein and Peter Borwein, Pi and the AGM, John Wiley & Sons, New York, 1987, Chap. 11.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Dirk Huylebrouck, Similarities in irrationality proofs for Pi, ln2, zeta(2) and zeta(3), Amer. Math. Monthly, Vol. 108, No. 3 (2001), pp. 222-231.
Jonathan Sondow, A faster product for pi and a new integral for ln(pi/2), The American Mathematical Monthly, Vol. 112, No. 8 (2005), pp. 729-734; Editor's endnotes, ibid., Vol. 113, No. 7 (2006), pp. 670-671; arXiv preprint, arXiv:math/0401406 [math.NT], 2004.

Cf. A019669, A094643.

Mathematica
```
RealDigits[ Log[Pi/2], 10, 111][[1]]
```

log(Pi/2) \\ Charles R Greathouse IV, Jun 23 2014

Equals Sum_{n>=1} zeta(2*n)/(n*2^(2*n)) (cf. Boros & Moll p. 131). - Jean-François Alcover, Apr 29 2013
Equals Re(log(log(I))). - Stanislav Sykora, May 09 2015
Equals Integral_{-oo..+oo} -log(1/2 + i*z)/cosh(Pi*z) dz, where i is the imaginary unit. - Peter Luschny, Apr 08 2018
Equals Integral_{0..Pi/2} (2/(Pi-2*t)-tan(t)) dt. - Clark Kimberling, Jul 10 2020
Equals -Sum_{k>=1} log(1 - 1/(2*k)^2). - Amiram Eldar, Aug 12 2020
Equals Sum_{k>=1} (-1)^(k+1) * log(1 + 1/k). - Amiram Eldar, Jun 26 2021
Equals A053510 - A002162. - R. J. Mathar, Jun 15 2023

A099974 Write log(2) as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

Original entry on oeis.org

1, 5, 13, 141, 653, 1677, 3725, 20109, 544397, 2641549, 6835853, 15224461, 32001677, 65556109, 132664973, 266882701, 803753613, 1877495437, 4024979085, 8319946381, 16909880973, 51269619341, 601025433229, 1700537061005

Offset: 0

N. J. A. Sloane, Nov 13 2004, based on correspondence from Artur Jasinski, Mar 25 2003

			log(2) = 0.69314718055994530941723212145817656807550013436025525412... = 0.1011000101110010000101111111011111010001110011110111100110101011110010011110... in binary.

Cf. A002162, A099969, A099970, A099971, A099972, A099973.

d = 100; l = First[RealDigits[N[Log[2], d], 2]]; Do[m = Take[l, n]; k = Length[m]; If[m[[k]] == 1, Print[FromDigits[Reverse[m], 2]]], {n, 1, d}] (* Ryan Propper, Aug 17 2005 *)
Module[{nn=50,l2},l2=RealDigits[Log[2],2,nn][[1]];Table[FromDigits[ Reverse[ Take[ l2,n]],2],{n,nn}]]//Union (* Harvey P. Dale, Mar 29 2016 *)

More terms from Ryan Propper, Aug 17 2005

A242023 Decimal expansion of Sum_{n >= 1} (-1)^(n + 1)24/(n(n + 1)(n + 2)(n + 3)).

Original entry on oeis.org

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

Offset: 0

Richard R. Forberg, Aug 11 2014

Sum of terms of the inverse of Binomial(n,4) or A000332, for n>=4, with alternating signs.
In general the sums of Binomial coefficients of this form appear to have the form  m*log(2) - r, where m is an integer and r is rational as below:
For Binomial(n,1):  m = 1, r = 0. See A002162.
For Binomial(n,2):  m = 4, r = 2. See A000217.
For Binomial(n,3):  m = 12 r = 15/2. See A000292.
For Binomial(n,4):  m = 32, r = 64/3. See A000332.
For Binomial(n,5):  m = 80, r = 655/12. See A000389.
For Binomial(n,6):  m = 192, r = 661/5. See A000579.
For Binomial(n,7):  m = 448, r = 9289/30. See A000580.
For Binomial(n,8):  m = 1024, r = 74432/105. See A000581.
This is generalized as follows:
m grows as A001787(k) = k*2^(k-1) for Binomial(n,k).
r * (k-1)! produces the integer sequence: a(k) = 0, 2, 15, 128, 1310, 15864, 222936, 3572736,  where a(k+1)/a(k) approaches 2*k for large k.
Results are precise to 100 digits or more using Mathematica.

			0.8473764445849165680180945...

G. C. Greubel, Table of n, a(n) for n = 0..5000

Cf. A000332, A000217, A000292, A242024, A002162, A001787.

[32*Log(2) - 64/3]; // G. C. Greubel, Nov 23 2017

Sum[N[(-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3)), 150], {n, 1, Infinity}]
RealDigits[32*Log[2] - 64/3, 10, 50][[1]] (* G. C. Greubel, Nov 23 2017 *)

32*log(2) - 64/3 \\ Michel Marcus, Aug 13 2014

sumalt(n=1, (-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3))) \\ Michel Marcus, Aug 14 2014

Equals 32*log(2) - 64/3.

Equals 32*(A259284-1). - R. J. Mathar, Jun 30 2021

A267316 Decimal expansion of the Dirichlet eta function at 5.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Jan 13 2016

			1/1^5 - 1/2^5 + 1/3^5 - 1/4^5 + 1/5^5 - 1/6^5 + ... = 0.972119770446909305935655143553469532553513362...

OEIS Wiki, Euler's alternating zeta function
Eric Weisstein's World of Mathematics, Dirichlet Eta Function
Wikipedia, Dirichlet Eta Function

Cf. A002162 (value at 1), A013663, A072691 (value at 2), A197070 (value at 3), A267315 (value at 4), A136676, A334604.

RealDigits[(15 Zeta[5])/16, 10, 120][[1]]

15*zeta(5)/16 \\ Michel Marcus, Feb 01 2016

s = RLF(0); s
RealField(110)(s)
for i in range(1, 10000): s += -((-1)^i/((i)^5))
print(s) # Terry D. Grant, Aug 05 2016

Equals Sum_{k > 0} (-1)^(k+1)/k^5 = (15*zeta(5))/16.

Equals Lim_{n -> infinity} A136676(n)/A334604(n). - Petros Hadjicostas, May 07 2020

A016639 Decimal expansion of log(16) = 4*log(2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.77258872223978123766892848583270627230200053744102101648272... - _Harry J. Smith_, May 17 2009

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Peter Bala, A continued fraction expansion for the constant log(16)
Eric Weisstein's World of Mathematics, Madelung Constants.
Index entries for transcendental numbers

Equals 4*A002162.
Equals (4/5)*A016655.
Equals A303658 + 2.
Cf. A016444 (continued fraction).
Cf. A001008, A002805, A188859.

Log(16); // Vincenzo Librandi, Feb 20 2015

RealDigits[Log[16], 10, 120][[1]] (* Harvey P. Dale, Jun 12 2012 *)

default(realprecision, 20080); x=log(16); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016639.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009, corrected May 19 2009

Equals 4*A002162.
Equals Sum_{k=1..4} (-1)^(k+1) gamma(0, k/4) where gamma(n,x) denotes the generalized Stieltjes constants. - Peter Luschny, May 16 2018
Equals -2 + Sum_{k>=1} H(k)*(k+1)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, May 28 2021
Equals 1 + Limit_{n -> infinity} (1/n)*Sum_{k = 1..n} (2*n + k)/(2*n - k) = 2*( 1 + Limit_{n -> infinity} (1/n)*Sum_{k = 1..n} (n - k)/(n + k) ). - Peter Bala, Oct 10 2021
Equals 2 + 1/(1 + 1/(3 + 2/(4 + 6/(5 + 6/(6 + 12/(7 + 12/(8 + ... + n*(n-1)/(2*n-1 + n*(n-1)/(2*n + ...))))))))). Cf. A188859. - Peter Bala, Mar 04 2024

A086054 Decimal expansion of Pi*log(2).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jul 07 2003

Madelung constant b2(2), negated.

			2.1775860903036021305006888982376139...

G. Boros and V. H. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, 2004 (equation 13.6.6).

Eric Weisstein's World of Mathematics, Madelung Constants

Cf. A000796 (Pi), A002162 (log(2)), A173623.

RealDigits[Pi Log[2],10,120][[1]] (* Harvey P. Dale, Dec 31 2011 *)

Pi*log(2) = -(8/3)*int(log(x)/sqrt(1+4*x-4*x^2), x=0..1). - John M. Campbell, Feb 07 2012
Pi*log(2) = int((x/sin(x))^2, x=0..Pi/2) = int(log(x^2+1)/(x^2+1), x=0..infinity) = int(-log(cos(x)), x=-Pi/2..Pi/2) = int(arctan(1/x)^2, x=0..infinity). - Jean-François Alcover, May 30 2013
From Amiram Eldar, Jul 11 2020: (Start)
Equals Integral_{x=-1..1} arcsin(x) dx / x.
Equals Integral_{x=-Pi/2..Pi/2} x*cot(x) dx. (End)
Equals Integral_{x = 0..oo} log(x^2 + 4)/(x^2 + 4) dx. - Peter Bala, Jul 22 2022
Equals -Im(Polylog(2, 2)). - Mohammed Yaseen, Jul 03 2024

Corrected by Antti Ahti (antti.ahti(AT)tkk.fi), Nov 17 2004

More terms from Benoit Cloitre, May 21 2005

A118858 Decimal expansion of log(2)/Pi^2.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, May 02 2006

			0.07023049277268287640...

Derrick Norman Lehmer, Asymptotic Evaluation of Certain Totient Sums, American Journal of Mathematics, Vol. 22, No. 4 (1900), pp. 293-335.
Eric Weisstein's World of Mathematics, Primitive Pythagorean Triple.

Cf. A249750, A328499, A002162, A002388, A284983.

RealDigits[Log[2]/Pi^2, 10, 100][[1]] (* Amiram Eldar, Jun 25 2021 *)

log(2)/Pi^2 \\ Michel Marcus, Jun 25 2021

Equals lim_{n->oo} A328499(n)/n. - Amiram Eldar, Jun 25 2021

Equals Integral_{z>=0} z*sech(Pi*z)^2. - Peter Luschny, Aug 03 2021

A187832 Decimal expansion of integral from 1/2 to 1 of (1-x)/x dx.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Dec 27 2012

Replacing 1/2 with any other number 0 < t < 1, the value of the integral is t - 1 - log(t).

			0.193147180559945309417232121458176568075500134360255254120680009493393621969...

J.-M. Monier, Cours, Analyse, Tome 4, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 4.3.14 pages 53 and 367.

Index entries for transcendental numbers.

Apart from the first digit the same as A002162.

Cf. A239354: Sum_{k>=1} 1/((2k)*(2k+1)*(2k+2)).

(evalf(log(2) - 1/2), 111); # Bernard Schott, Nov 25 2021

Mathematica
```
RealDigits[Log[2] - 1/2, 10, 111][[1]]
```

log(2)-1/2 \\ Charles R Greathouse IV, Dec 27 2012

Equals log(2) - 1/2 = A002162 - 1/2.
Equals Sum_{k>=1} 1/((2k-1)*(2k)*(2k+1)). - Bruno Berselli, Mar 16 2014
From Amiram Eldar, Jul 28 2020: (Start)
Equals Sum_{k>=0} (-1)^k/(k+3).
Equals Sum_{k>=2} 1/(k * 2^k).
Equals Sum_{k>=2} 1/(4*k^2 - 2*k).
Equals Sum_{k>=2} (zeta(k) - 1)/2^k.
Equals Sum_{k>=1} zeta(2*k + 1)/2^(2*k + 1). (End)
From Bernard Schott, Nov 22 2021: (Start)
Equals Sum_{k>=1} (S(k) - log(2)) when S(k) = Sum_{m=1..k} (-1)^(m+1) / m.
Equals Integral_{x=0..1} x/(1+x)^2 dx. (End)
Equals Sum_{k,m>=1} (-1)^(k+m)/(k+m). - Amiram Eldar, Jun 09 2022
Equals Integral_{x = 0..1} Integral_{y = 0..1} x*y/(x + y)^2 dy dx. - Peter Bala, Dec 12 2022

cp's OEIS Frontend

A016578 Decimal expansion of log(3/2).

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A016730 Continued fraction for log(2).

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A094642 Decimal expansion of log(Pi/2).

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A099974 Write log(2) as a binary fraction; read this from left to right and whenever a 1 appears, note the integer formed by reading leftwards from that 1.

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A242023 Decimal expansion of Sum_{n >= 1} (-1)^(n + 1)*24/(n*(n + 1)*(n + 2)*(n + 3)).

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Magma

Mathematica

PARI

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A267316 Decimal expansion of the Dirichlet eta function at 5.

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Sage

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A016639 Decimal expansion of log(16) = 4*log(2).

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A242023 Decimal expansion of Sum_{n >= 1} (-1)^(n + 1)24/(n(n + 1)(n + 2)(n + 3)).