A016628 -id:A016628 - cp's OEIS frontend

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

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)).

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))).

A016638 Decimal expansion of log(15).

Original entry on oeis.org

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

A016648 Decimal expansion of log(25).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.218875824868200749201518666452375279051202708537035443825295782948357....

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. A000045, A016453 (continued fraction), A016628.

RealDigits[Log[25], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(25); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016648.txt", n, " ", d)); \\ Harry J. Smith, May 19 2009

Equals 10 * Sum_{k>=1} F(k)^2/(k * 4^k), where F(k) is the k-th Fibonacci number (A000045). - Amiram Eldar, Aug 09 2020

Equals 2*A016628. - Alois P. Heinz, Aug 07 2023

A016579 Decimal expansion of log(5/2).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane

			0.916290731874155065183527211768011071450101219908262467791967881980785...

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

Cf. A016530 (continued fraction).

SetDefaultRealField(RealField(100)); Log(5/2); // Vincenzo Librandi, Apr 07 2020

RealDigits[Log[5/2], 10, 120][[1]] (* Vincenzo Librandi, Apr 07 2020 *)

default(realprecision, 20080); x=10*log(5/2); for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b016579.txt", n, " ", d)); \\ Harry J. Smith, May 25 2009

Equals A016628 minus A002162. - R. J. Mathar, Jan 23 2009

log(5/2) = 2*Sum_{n >= 1} 1/(n*P(n, 7/3)*P(n-1, 7/3)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(5/2) = 0.916290731874155065183527(19...), correct to 24 decimal places. - Peter Bala, Mar 18 2024

Leading zero removed and offset adjusted by R. J. Mathar, Feb 06 2009

A016668 Decimal expansion of log(45).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.806662489770319757391249807071239048820582469914016625382036558749167....

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. A016473 (continued fraction).

RealDigits[Log[45], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(45); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016668.txt", n, " ", d)); \\ Harry J. Smith, May 21 2009

Equals 2*A002391 + A016628. - R. J. Mathar, Jun 07 2024

A016673 Decimal expansion of log(50).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.912023005428146058618750787910551847126702842897290697945975792441751....

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. A016478 (continued fraction).

RealDigits[Log[50], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(50); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016673.txt", n, " ", d)); \\ Harry J. Smith, May 21 2009

Equals A002162 + 2*A016628. - R. J. Mathar, Jun 07 2024

A016678 Decimal expansion of log(55).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			4.007333185232470918662702911191316939347308208205934897131215600604752....

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. A016483 (continued fraction).

RealDigits[Log[55], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(55); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016678.txt", n, " ", d)); \\ Harry J. Smith, May 22 2009

Equals A016628+A016634. - R. J. Mathar, Jun 07 2024

A016683 Decimal expansion of log(60).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			4.094344562222100684830468813065066480324092180811777681888702244098460....

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. A016488 (continued fraction).

RealDigits[Log[60], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(60); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016683.txt", n, " ", d)); \\ Harry J. Smith, May 22 2009

Equals 2*A002162+A002391+A016628. - R. J. Mathar, Jul 22 2025

A068434 Expansion of log(5) in base 2.

Original entry on oeis.org

1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1

Offset: 1

Benoit Cloitre, Mar 09 2002

Iain Fox, Table of n, a(n) for n = 1..20000

Cf. A016628.

RealDigits[Log[5], 2, 125][[1]] (* Alonso del Arte, Dec 14 2017 *)

concat(binary(log(5))) \\ Michel Marcus, Dec 14 2017

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A016638 Decimal expansion of log(15).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A016648 Decimal expansion of log(25).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A016579 Decimal expansion of log(5/2).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A016668 Decimal expansion of log(45).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A016673 Decimal expansion of log(50).

Views

Author

Keywords

Examples

References