A002391 -id:A002391 - cp's OEIS frontend

A016644 Decimal expansion of log(21).

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

Offset: 1

N. J. A. Sloane

			3.044522437723422996500597980365705434284575287404610640194084483575074....

Milton Abramowitz and Irene 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
Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Index entries for transcendental numbers

Cf. A016449 (continued fraction).

RealDigits[Log[21], 10, 120][[1]] (* Harvey P. Dale, Sep 04 2012 *)

default(realprecision, 20080); x=log(21); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016644.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009, corrected May 20 2009

Eqals A016630 + A002391. - R. J. Mathar, Jul 22 2025

A175475 Decimal expansion of the Dickman function evaluated at 1/3.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, May 25 2010

Density of the cube root-smooth numbers, see A090081. - Charles R Greathouse IV, Jul 14 2014

			F(1/3) = 0.04860838829113156690718...

David Broadhurst, Dickman polylogarithms and their constants arXiv:1004.0519 [math-ph], 2010.
K. Soundararajan, An asymptotic expansion related to the Dickman function, arXiv:1005.3494 [math.NT], 2010.

Cf. A002391, A072691, A175478.

N[1 - Log[3] + Log[3]^2/2 - Pi^2/12 + PolyLog[2, 1/3], 105] // RealDigits // First // Prepend[#, 0]& (* Jean-François Alcover, Feb 05 2013 *)

1-log(3)+log(3)^2/2-Pi^2/12+polylog(2,1/3) \\ Charles R Greathouse IV, Jul 14 2014

Equals 1 - log(3) + log^2(3)/2 - Pi^2/12 + Sum_{n>=1} 1/(n^2*3^n), where Sum_{n>=1} 1/(n^2*3^n) = 0.3662132299770634876167462976642627638...

A191907 Square array read by antidiagonals up: T(n,k) = -(n-1) if n divides k, else 1.

Original entry on oeis.org

0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 1, 1, -2, -1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, -3, 1, -1, 0, 1, 1, 1, 1, 1, -2, 1, 0, 1, 1, 1, 1, -4, 1, 1, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, -5, 1, -3, -2, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, -6, 1, 1, 1, 1, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, -4, 1, -2, 1, 0, 1, 1, 1, 1, 1, 1, 1, -7, 1, 1, 1, -3, 1, -1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Mats Granvik, Jun 19 2011

Apart from the top row, the same as A177121.

Sum_{k>=1} T(n,k)/k = log(n); this has been pointed out by Jaume Oliver Lafont in A061347 and A002162.

			Table starts:
0..0..0..0..0..0..0..0..0...
1.-1..1.-1..1.-1..1.-1..1...
1..1.-2..1..1.-2..1..1.-2...
1..1..1.-3..1..1..1.-3..1...
1..1..1..1.-4..1..1..1..1...
1..1..1..1..1.-5..1..1..1...
1..1..1..1..1..1.-6..1..1...
1..1..1..1..1..1..1.-7..1...
1..1..1..1..1..1..1..1.-8...

Cf. A002162, A002391, A016627, A191904.

Clear[t, n, k];
nn = 30;
t[n_, k_] := t[n, k] = If[Mod[n, k] == 0, -(k - 1), 1]
MatrixForm[Transpose[Table[Table[t[n, k], {k, 1, nn}], {n, 1, nn}]]]

N=20; M=matrix(N,N,n,k, if(n%k==0,1-k,1))~

If n divides k then T(n,k) = -(n-1) else 1.

A322334 Factorial expansion of log(3) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 0, 0, 2, 1, 5, 0, 0, 0, 4, 3, 0, 0, 9, 6, 1, 11, 4, 13, 8, 9, 0, 16, 2, 14, 24, 9, 22, 5, 26, 4, 2, 31, 15, 17, 15, 8, 31, 18, 17, 20, 36, 20, 3, 41, 12, 7, 44, 44, 2, 38, 20, 44, 47, 3, 44, 19, 40, 9, 14, 1, 24, 15, 46, 0, 60, 37, 67, 63, 24, 64, 51, 30, 31, 59, 18, 68, 63, 22, 16, 45, 29, 43, 24, 13, 26, 77, 30, 37, 41, 3, 29, 25, 88, 12, 93, 56, 60, 60, 13

Offset: 1

G. C. Greubel, Dec 03 2018

nonn

			log(3) = 1 + 0/2! + 0/3! + 2/4! + 1/5! + 5/6! + 0/7! + 0/8! + ...

Cf. A002391 (decimal expansion), A016731 (continued fraction).

Cf. A067882 (log(2)), A322333 (log(5)), A068460 (log(7)), A068461 (log(11)).

SetDefaultRealField(RealField(250));  [Floor(Log(3))] cat [Floor(Factorial(n)*Log(3)) - n*Floor(Factorial((n-1))*Log(3)) : n in [2..80]];

With[{b = Log[3]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = log(3); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

def a(n):
    if (n==1): return floor(log(3))
    else: return expand(floor(factorial(n)*log(3)) - n*floor(factorial(n-1)*log(3)))
[a(n) for n in (1..80)]

A016731 Continued fraction for log(3).

Original entry on oeis.org

1, 10, 7, 9, 2, 2, 1, 3, 1, 32, 2, 17, 1, 15, 1, 1, 7, 3, 1, 35, 1, 1, 1, 2, 5, 3, 2, 1, 4, 2, 1, 3, 1, 5, 3, 13, 1, 1, 1, 6, 2, 3, 1, 152, 1, 2, 3, 1, 7, 9, 2, 1, 3, 19, 2, 2, 2, 3, 2, 5, 1, 1, 4, 1, 19, 5, 4, 2, 1, 2, 7, 4, 2, 1, 6, 3, 2, 2, 4, 1, 1, 1, 4, 1

Offset: 0

N. J. A. Sloane

			1.09861228866810969139524523... = 1 + 1/(10 + 1/(7 + 1/(9 + 1/(2 + ...)))). - _Harry J. Smith_, May 16 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
G. Xiao, Contfrac

Cf. A002391 (decimal expansion).

ContinuedFraction(Log(3)); // G. C. Greubel, Sep 15 2018

ContinuedFraction[Log[3], 100] (* G. C. Greubel, Sep 15 2018 *)

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(log(3)); for (n=1, 20000, write("b016731.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 16 2009

Offset changed by Andrew Howroyd, Jul 10 2024

A059543 Beatty sequence for log(3).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96

Offset: 1

Mitch Harris, Jan 22 2001

Differs from A160542 at indices n=81, 91, 101, 111, 121, 131, 141, 151, 152, 161 etc. - R. J. Mathar, May 20 2009

Harry J. Smith, Table of n, a(n) for n = 1..2000
Aviezri S. Fraenkel, Jonathan Levitt, 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 A059544.
Cf. A002391 (log(3)).
Cf. A160542.

A059543 := proc(n)
    floor(n*log(3)) ;
end proc:
seq(A059543(n),n=1..100) ; # R. J. Mathar, Jun 26 2023

Floor[Range[100]*Log[3]] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=log(3); for (n = 1, 2000, write("b059543.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*A002391). - Paolo Xausa, Jul 05 2024

A113209 Decimal expansion of log(5)/log(3).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Oct 17 2005

Capacity dimension of the box fractal.

Hausdorff dimension of the graph of Bourbaki's function. McCollum: We examine Bourbaki's function, an easily-constructed continuous but nowhere-differentiable function, and explore properties including functional identities, the antiderivative, and the Hausdorff dimension of the graph. - Jonathan Vos Post, Sep 15 2010

			1.4649735207179...

James McCollum, Properties of Bourbaki's Function, Sep 15, 2010.
Eric Weisstein's World of Mathematics, Box Fractal
Index entries for transcendental numbers

Cf. A152914.

Digits:=100: evalf(log(5)/log(3)); # Wesley Ivan Hurt, Jul 07 2014

RealDigits[Log[3, 5], 10, 100][[1]] (* Alonso del Arte, Jul 07 2014 *)

log(5)/log(3) \\ Charles R Greathouse IV, May 15 2019

Equals A016628 divided by A002391. - R. J. Mathar, Sep 08 2013

A145960 Decimal expansion of 2*log(5/3) used in BBP Pi formula.

Original entry on oeis.org

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

Offset: 1

Artur Jasinski, Oct 25 2008

BBP formula for Pi = 4*A145963 - (1/2)*A145960 - (1/2)*(A145961-A145962).

			1.021651247531981366411...

Eric Weisstein's World of Mathematics, BBP Formula
Index entries for transcendental numbers

Cf. A000796, A145961, A145962, A145963.

Mathematica
```
First[RealDigits[2 Log[5/3], 10, 100]]
```

2*log(5/3) \\ Michel Marcus, Apr 05 2015

Equals 2*log(5/3) = 2*(log(5)-log(3)) = 2*(A016628 - A002391) = log(25/9) = 4*arctanh(1/4).
Equals Hypergeometric2F1(1, 1/2, 3/2, 1/16).
Equals Sum_{k>=0} (1/16)^k*(1/(2k+1)).

A254619 a(n) = 4^n(2n + 1)!/n!.

Original entry on oeis.org

1, 24, 960, 53760, 3870720, 340623360, 35424829440, 4250979532800, 578133216460800, 87876248902041600, 14763209815542988800, 2716430606059909939200, 543286121211981987840000, 117349802181788109373440000

Offset: 0

Peter Bala, Feb 03 2015

Cf. A002391, A034910, A073000, A254381, A254620.

Maple
```
seq(4^n*(2*n + 1)!/n!, n = 0..13);
```

Table[4^n (2n+1)!/n!,{n,0,20}] (* Harvey P. Dale, Oct 02 2021 *)

E.g.f.: 1/(1 - 16*x)^(3/2) = 1 + 24*x + 960*x^2/2! + 53760*x^3/3! + ....
Recurrence equation: a(n) = 8*(2*n + 1)*a(n-1) with a(0) = 1.
2nd order recurrence equation: a(n) = (20*n + 6)*a(n-1) - 16*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 24.
Define a sequence b(n) := a(n)*sum {k = 0..n} 1/((2*k + 1)*4^k) beginning [1, 26, 1052, 59032, 4251984, 374204832, 38917967808, ...]. It is not difficult to check that b(n) also satisfies the previous 2nd order recurrence equation (and so is an integer sequence). From this observation we can obtain the continued fraction expansion
log(3) = Sum {k >= 0} 1/((2*k + 1)*4^k) = 1 + 2/(24 - 16*3^2/(46 - 16*5^2/(66 - ... - 16*(2*n - 1)^2/((20*n + 6) - ... )))).
Alternative 2nd order recurrence equation: a(n) = (12*n + 10)*a(n-1) + 16*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 24.
Define now a sequence c(n) := a(n)*sum {k = 0..n} (-1)^k/((2*k + 1)*4^k) beginning [1, 22, 892, 49832, 3589584, 315853152, 32849393088, ...], which, along with a(n), satisfies the alternative 2nd order recurrence equation. From this observation we find the continued fraction expansion 2*arctan(1/2) = Sum {k >= 0} (-1)^k/((2*k + 1)*4^k) = 1 - 2/(24 + 16*3^2/(34 + 16*5^2/(46 + ... + 16*(2*n - 1)^2/((12*n + 10) + ... )))). Cf. A254381 and A254620.

A379324 Decimal expansion of log(6)^(log(5)^(log(4)^(log(3)^log(2)))).

Original entry on oeis.org

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

Offset: 1

Paolo Xausa, Dec 20 2024

This is an approximation to Pi accurate to 5 digits.

			3.1415773871691905336574449813486768105453105619...

Paolo Xausa, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pi Approximations.
Index entries for sequences related to the number Pi

Cf. A000796, A002162, A002391, A016627, A016628, A016629.

Cf. A221185, A379276, A379321, A379322, A379323.

First[RealDigits[Log[6]^Log[5]^Log[4]^Log[3]^Log[2], 10, 100]]

Equals A016629^(A016628^(A016627^(A002391^A002162))).

cp's OEIS Frontend

A016644 Decimal expansion of log(21).

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A175475 Decimal expansion of the Dickman function evaluated at 1/3.

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A191907 Square array read by antidiagonals up: T(n,k) = -(n-1) if n divides k, else 1.

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A322334 Factorial expansion of log(3) = Sum_{n>=1} a(n)/n!.

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A016731 Continued fraction for log(3).

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A059543 Beatty sequence for log(3).

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A113209 Decimal expansion of log(5)/log(3).

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A254619 a(n) = 4^n(2n + 1)!/n!.