A002391 -id:A002391 - cp's OEIS frontend

A356581 Decimal expansion of gamma - 3*log(2) + log(3) + 17/24.

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

Offset: 0

Amiram Eldar, Aug 13 2022

			0.30471974522313995708339429596373176479648282401524...

Chao-Ping Chen and H. M. Srivastava, New representations for the Lugo and Euler-Mascheroni constants. II, Applied Mathematics Letters, Vol. 25, No. 3 (2012), pp. 333-338.

Cf. A001620 (gamma), A002162, A002391, A356580.

RealDigits[EulerGamma - 3*Log[2] + Log[3] + 17/24, 10, 100][[1]]

PARI
```
Euler - 3*log(2) + log(3) + 17/24
```

Equals lim_{n->oo} Sum_{i,j,k=1..n} 1/(i+j+k) - log(n) - 9*(log(2) - log(3)/2)*n + 3*(3*log(3)/2 - 2*log(2))*n^2 (Chen and Srivastava, 2012).

A365522 Decimal expansion of (Pisqrt(3) + 9log(3))/24.

Original entry on oeis.org

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

Offset: 0

Claude H. R. Dequatre, Sep 08 2023

This sequence is also the decimal expansion of Sum_{k>=1} 1/(f(k) +g(k)), where f(k) and g(k) are respectively the k-th triangular and the 13-gonal numbers (A000217 and A051865).

			0.63870452877981836559747674605121660577831724019512...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Ian Shamos, A catalog of the real numbers (2011), p. 544.
Wikipedia, Polygonal number.

Cf. A000217, A000796, A002194, A002391, A051865.

Cf. A244639, A244641, A244645, A244646, A244647, A244648, A244649.

SetDefaultRealField(RealField(139)); R:= RealField(); (Pi(R)*Sqrt(3)+9*Log(3))/24; // G. C. Greubel, Mar 24 2024

RealDigits[(Pi*Sqrt[3] + 9*Log[3])/24, 10 , 100][[1]] (* Amiram Eldar, Sep 08 2023 *)

PARI
```
(Pi*sqrt(3)+9*log(3))/24
```

numerical_approx((pi*sqrt(3)+9*log(3))/24, digits=139) # G. C. Greubel, Mar 24 2024

Equals Sum_{k>=1} 1/(6*k^2 - 4*k) = A244645/2 [Shamos].

Equals - Integral_{x=0..1} log(1-x^6)/x^5 dx [Shamos].

A145422 Decimal expansion of sum_{n=0..infinity} (-1)^n/(2^(3n)*(3n+1)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 08 2009

			0.97080388430077584732977...

Alexander Apelblat, Tables of Integrals and Series, Harri Deutsch, (1996), 4.1.17

Maple
```
evalf((Pi/sqrt(3)+log(3))/3) ;
```

RealDigits[Hypergeometric2F1[1/3,1,4/3,-(1/8)],10,120][[1]] (* Harvey P. Dale, May 11 2017 *)

(A093602+A002391)/3 .

A157718 Greedy Egyptian fraction expansion of log(3).

Original entry on oeis.org

1, 11, 130, 91827, 42593758221, 2068726045016880942060, 20697114911379630588051784011292634933847536, 832769470129253476302780470023395858447487389073547955500158020204885523374048803963217

Offset: 0

Jaume Oliver Lafont, Mar 04 2009

			log(3) = Sum_{n>=0} 1/a(n) = 1/1 + 1/11 + 1/130 + 1/91827 + 1/42593758221 + ...

Wikipedia, Greedy algorithm for Egyptian fractions

Cf. A058962, A154920, A157024, A002391, A118324.

x=log(3); for (k=1, 8, d=ceil(1/x); x=x-1/d; print(d,","))

A293079 Decimal expansion of log_Pi(3).

Original entry on oeis.org

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

Offset: 0

Rajat Goel, Sep 30 2017

			0.959713118569390019336023198...

Cf. A000796, A002391, A053510, A104288.

RealDigits[Log[Pi,3],10,120][[1]] (* Harvey P. Dale, Apr 09 2021 *)

log(3)/log(Pi) \\ Michel Marcus, Sep 30 2017

from math import log, pi
print(log(3,pi))

Equals log(3)/log(Pi) = A002391/A053510.

A322758 Decimal expansion of log(8/sqrt(3)).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 28 2018

Equals 3*A002162 - (1/2)*A002391. - Jianing Song, Dec 29 2018

			1.530135397345781082554073745913266851902755124169391036494692861661433719...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 557.

Index entries for transcendental numbers

Cf. A002162, A002391.

RealDigits[Log[8/Sqrt[3]],10,120][[1]] (* Harvey P. Dale, Aug 31 2021 *)

log(8/sqrt(3)) \\ Charles R Greathouse IV, May 15 2019

A323458 Decimal expansion of log(2^(1/2)*3^(1/3) / 6^(1/6)).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Jan 20 2019

			0.4141511082980000517049515799731464734664151377572...

Index entries for transcendental numbers

Suggested by A230191.
Cf. A002162, A002391, A016635.
Cf. A001008, A002805.

RealDigits[Log[2^(1/2)*3^(1/3) / 6^(1/6)], 10, 101][[1]] (* Georg Fischer, Apr 04 2020 *)

log( 2^(1/2)*3^(1/3) / 6^(1/6) ) \\ Charles R Greathouse IV, May 15 2019

From Jianing Song, Jan 23 2019: (Start)
Equals (1/6)*log(12) = (1/6)*A016635.
Equals (1/3)*log(2) + (1/6)*log(3) = (1/3)*A002162 + (1/6)*A002391. (End)
Equals Sum_{k>=1} H(2*k-1)/4^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. - Amiram Eldar, May 30 2021

a(99) corrected by Georg Fischer, Apr 04 2020

A373214 Signature sequence of log(3).

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 13, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 14, 2, 13, 1, 12, 11, 10, 9, 8, 7, 6, 5, 4, 15, 3, 14, 2, 13, 1, 12, 11, 10, 9, 8, 7, 6, 5, 16, 4, 15, 3, 14, 2, 13, 1, 12, 11, 10

Offset: 1

R. J. Mathar, May 28 2024

nonn

Signature sequence of x = A002391: defined by sorting the values of i+j*x, i,j>=1, and collecting the list of the i in that order.

Starts similar to A004736, because log(3)=1.09861... is close to 1.

Index entries for sequences related to signature sequences

Cf. A002391.

A387247 Decimal expansion of (2*log(3) + 7)/8.

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, Aug 24 2025

			1.149653072167027422848811309230631426161872639...

W. Lai, H. Jiang, D. Zhu, and B. Zhu, Improved Approximation Algorithm and Hardness Result for Sorting Unsigned Strings by Symmetric Reversals. In: Fomin, F.V., Xiao, M. (eds) Computing and Combinatorics. COCOON 2025. Lecture Notes in Computer Science, vol 15984. Springer, Singapore (2026).
Index entries for transcendental numbers.

Cf. A002391, A016632.

Mathematica
```
RealDigits[(2Log[3]+7)/8,10,100][[1]]
```

cp's OEIS Frontend

A356581 Decimal expansion of gamma - 3*log(2) + log(3) + 17/24.

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A365522 Decimal expansion of (Pi*sqrt(3) + 9*log(3))/24.

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A145422 Decimal expansion of sum_{n=0..infinity} (-1)^n/(2^(3n)*(3n+1)).

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A157718 Greedy Egyptian fraction expansion of log(3).

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A293079 Decimal expansion of log_Pi(3).

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A322758 Decimal expansion of log(8/sqrt(3)).

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A323458 Decimal expansion of log(2^(1/2)*3^(1/3) / 6^(1/6)).

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A365522 Decimal expansion of (Pisqrt(3) + 9log(3))/24.