A002391 -id:A002391 - cp's OEIS frontend

A016638 Decimal expansion of log(15).

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

Offset: 1

N. J. A. Sloane

			2.708050201102210065996004570148713344173091912091267173647342225111673....

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].
Index entries for transcendental numbers

Cf. A016443 (continued fraction).

RealDigits[Log[15], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(15); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016638.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009

Equals A002391 + A016628. - R. J. Mathar, Jun 10 2024

Final digits of sequence corrected using the b-file. - N. J. A. Sloane, Aug 30 2009

A016641 Decimal expansion of log(18).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.890371757896164692207722595303227977370481250005754157590068676768382....

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].
Index entries for transcendental numbers

Cf. A016446 (continued fraction).

RealDigits[Log[18], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(18); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016641.txt", n, " ", d)); \\ Harry J. Smith, May 17 2009

Equals A002162 + 2*A002391. - R. J. Mathar, Jun 10 2024

A016647 Decimal expansion of log(24).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.178053830347945619646941601297055408873990960903515214096734362117675...

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].
Index entries for transcendental numbers

Cf. A016452 Continued fraction.

RealDigits[Log[24],10,120][[1]] (* Harvey P. Dale, Oct 04 2021 *)

default(realprecision, 20080); x=log(24); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016647.txt", n, " ", d)); \\ Harry J. Smith, May 19 2009

Equals 3*A002162 +A002391. - R. J. Mathar, Jul 22 2025

A059548 Beatty sequence for 1 + log(3).

Original entry on oeis.org

2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 128, 130, 132, 134

Offset: 1

Mitch Harris, Jan 22 2001

Different from A064720, although the sequences agree for about 1400 terms. - Joshua Zucker, Apr 25 2007

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 A059547.

Cf. A002391.

Floor[Range[100]*(1 + Log[3])] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=1 + log(3); for (n = 1, 2000, write("b059548.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*(1 + A002391)). - Paolo Xausa, Jul 05 2024

A156057 Decimal expansion of log(3)/2.

Original entry on oeis.org

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

Offset: 0

Jonathan Vos Post, Feb 03 2009

Culler & Shalen show a bound of log(3)/2 on maximal injectivity under certain circumstances, see links.

Equals arctanh(1/2), the rapidity of an object traveling at half the speed of light. - Sean Stroud, May 13 2019

			0.54930614433405484569762261846...

Marc Culler and Peter B. Shalen, Betti numbers and injectivity radii, Proceedings of the American Mathematical Society, Vol. 137, No. 11 (2009), pp. 3919-3922; preprint, arXiv:0902.0014 [math.GT], 2009.
R. S. Melham and A. G. Shannon, Inverse Trigonometric Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
Michael Penn, This gnarly integral is actually easy??, YouTube video, 2023.
Index entries for transcendental numbers

Cf. A000045, A002391 (decimal expansion of natural logarithm of 3).

RealDigits[Log[3]/2,10,120][[1]] (* Harvey P. Dale, Apr 13 2016 *)

log(3)/2 \\ Charles R Greathouse IV, May 15 2019

Equals arctanh(1/2) = arccoth(2) = Integral_{x>2} 1/(x^2-1) dx. - Jean-François Alcover, Jun 04 2013
From Amiram Eldar, Aug 05 2020: (Start)
Equals Sum_{k>=0} 1/((2*k+1) * 2^(2*k+1)).
Equals Integral_{x=0..oo} 1/(exp(x) + 2) dx. (End)
Equals Sum_{k>=1} arctanh(1/Fibonacci(2*k+2)) (Melham and Shannon, 1995). - Amiram Eldar, Jan 15 2022
log(3)/2 = Sum_{n >= 1} 1/(n*P(n, 2)*P(n-1, 2)), where P(n, x) denotes the n-th Legendre polynomial. The first 10 terms of the series gives the approximation log(3)/2 = 0.54930614433(10...), correct to 11 decimal places. - Peter Bala, Mar 16 2024

All digits were wrong. Corrected by N. J. A. Sloane, Feb 05 2009

Offset 0 from Michel Marcus, May 13 2019

A175478 Decimal expansion of log(3)^2.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, May 25 2010

			1.2069489608125819778437...

Index entries for transcendental numbers

Cf. A002391, A253191.

RealDigits[Log[3]^2,10,120][[1]] (* Harvey P. Dale, Jan 23 2012 *)

log(3)^2 \\ Charles R Greathouse IV, May 15 2019

Equals A002391^2.

Equals Sum_{n >= 0} (-1)^n*(4/3)^(n+1)/((n+1)^2*binomial(2*n+1,n)). See my entry in A002544 dated Apr 18 2017. - Peter Bala, Jan 30 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).

A254381 a(n) = 3^n(2n + 1)!/n!.

Original entry on oeis.org

1, 18, 540, 22680, 1224720, 80831520, 6304858560, 567437270400, 57878601580800, 6598160580211200, 831368233106611200, 114728816168712345600, 17209322425306851840000, 2787910232899709998080000, 485096380524549539665920000, 90227926777566214377861120000

Offset: 0

Peter Bala, Feb 04 2015

Cf. A002391, A034910, A073000, A093766, A254619, A254620.

Maple
```
seq(3^n*(2*n + 1)!/n!, n = 0..13);
```

Table[3^n(2n + 1)!/n!, {n, 0, 19}] (* Alonso del Arte, Feb 04 2015 *)

E.g.f.: 1/(1 - 12*x)^(3/2) = 1 + 18*x + 540*x^2/2! + 22680*x^3/3! + ....
Recurrence equation: a(n) = 6*(2*n + 1)*a(n-1) with a(0) = 1.
2nd order recurrence equation: a(n) = 8*(n + 1)*a(n-1) + 12*(2*n - 1)^2*a(n-2) with a(0) = 1, a(1) = 18.
Define a sequence b(n) := a(n)*sum {k = 0..n} (-1)^k/((2*k + 1)*3^k) beginning [1, 16, 492, 20544, 1111056, 73299456, 5718022848, ...]. It is not difficult to check that b(n) also satisfies the previous 2nd order recurrence equation (and so is an integer sequence). Using this observation we obtain the continued fraction expansion Pi/(2*sqrt(3)) = Sum {k >= 0} (-1)^k/( (2*k + 1)*3^k ) = 1 - 2/(18 + 12*3^2/(24 + 12*5^2/(32 + ... + 12*(2*n - 1)^2/((8*n + 8) + ... )))). Cf. A254619 and A254620.

A293383 Decimal expansion of Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Oct 13 2017

This constant plus A293384 equals log(3), due to the identity:

Sum_{n=-oo..+oo, n<>0} (x - y^n)^n / n = -log(1-x), here x = 2/3, y = 1/2.

			Constant t = 0.3927715755550675118591118772612280913427234490422634862023883438....
such that
t = (2^2 - 3)/(1*3*2) + (2^3 - 3)^2/(2*3^2*2^4) + (2^4 - 3)^3/(3*3^3*2^9) + (2^5 - 3)^4/(4*3^4*2^16) + (2^6 - 3)^5/(5*3^5*2^25) + (2^7 - 3)^6/(6*3^6*2^36) + (2^8 - 3)^7/(7*3^7*2^49) + (2^9 - 3)^8/(8*3^8*2^64) + (2^10 - 3)^9/(9*3^9*2^81) +...+ (2^(n+1) - 3)^n/(n * 3^n * 2^(n^2)) +...
More explicitly,
t = 1/(1*3*2) + 5^2/(2*9*2^4) + 13^3/(3*27*2^9) + 29^4/(4*81*2^16) + 61^5/(5*243*2^25) + 125^6/(6*729*2^36) + 253^7/(7*2187*2^49) + 509^8/(8*6561*2^64) + 1021^9/(9*19683*2^81) + 2045^10/(10*59049*2^100) + 4093^11/(11*177147*2^121) + 8189^12/(12*531441*2^144) +...
Also,
log(3) - t = 3/(1*2*(3-1)) - 3^2/(2*4*(3*2-1)^2) + 3^3/(3*8*(3*2^2-1)^3) - 3^4/(4*16*(3*2^3-1)^4) + 3^5/(5*32*(3*2^4-1)^5) - 3^6/(6*64*(3*2^5-1)^6) + 3^7/(7*128*(3*2^6-1)^7) +...+ -(-1)^n*3^n/(n*2^n*(3*2^(n-1) - 1)^n) +...

Cf. A293384, A293381, A293382, A292178, A292179, A002391 (log 3).

{t = suminf(n=1, 1.*(2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)) )}
for(n=1,120, print1(floor(10^n*t)%10,", "))

Constant: Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

Constant: log(3) - Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).

A293384 Decimal expansion of Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Oct 13 2017

This constant plus A293383 equals log(3), due to the identity:

Sum_{n=-oo..+oo, n<>0} (x - y^n)^n / n = -log(1-x), here x = 2/3, y = 1/2.

			Constant t = 0.7058407131130421795361333596612976133047671087804859655323059...
such that
t = 3/(1*2*(3-1)) - 3^2/(2*4*(3*2-1)^2) + 3^3/(3*8*(3*2^2-1)^3) - 3^4/(4*16*(3*2^3-1)^4) + 3^5/(5*32*(3*2^4-1)^5) - 3^6/(6*64*(3*2^5-1)^6) + 3^7/(7*128*(3*2^6-1)^7) - 3^8/(8*256*(3*2^7-1)^8) +...+ -(-1)^n*3^n/(n*2^n*(3*2^(n-1) - 1)^n) +...
More explicitly,
t = 3/(1*2*2) - 9/(2*4*5^2) + 27/(3*8*11^3) - 81/(4*16*23^4) + 243/(5*32*47^5) - 729/(6*64*95^6) + 2187/(7*128*191^7) - 6561/(8*256*383^8) + 19683/(9*512*767^9) - 59049/(10*1024*1535^10) + 177147/(11*2048*3071^11) - 531441/(12*4096*6143^12) +...
Also,
log(3) - t = (2^2 - 3)/(1*3*2) + (2^3 - 3)^2/(2*3^2*2^4) + (2^4 - 3)^3/(3*3^3*2^9) + (2^5 - 3)^4/(4*3^4*2^16) + (2^6 - 3)^5/(5*3^5*2^25) + (2^7 - 3)^6/(6*3^6*2^36) + (2^8 - 3)^7/(7*3^7*2^49) + (2^9 - 3)^8/(8*3^8*2^64) + (2^10 - 3)^9/(9*3^9*2^81) +...+ (2^(n+1) - 3)^n/(n * 3^n * 2^(n^2)) +...

Cf. A293383, A293381, A293382, A292178, A292179, A002391 (log 3).

{t = suminf(n=1, -1.*(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n) )}
for(n=1,120, print1(floor(10^n*t)%10,", "))

Constant: Sum_{n>=1} -(-1)^n * 3^n / (n * 2^n * (3*2^(n-1) - 1)^n).

Constant: log(3) - Sum_{n>=1} (2^(n+1) - 3)^n / (n * 3^n * 2^(n^2)).

cp's OEIS Frontend

A016638 Decimal expansion of log(15).

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A016641 Decimal expansion of log(18).

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A016647 Decimal expansion of log(24).

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A059548 Beatty sequence for 1 + log(3).

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A156057 Decimal expansion of log(3)/2.

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A175478 Decimal expansion of log(3)^2.

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

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A254381 a(n) = 3^n(2n + 1)!/n!.