A002391 -id:A002391 - cp's OEIS frontend

A386734 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x+y+z)} dx dy dz, where {} denotes fractional part.

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

Offset: 0

Amiram Eldar, Aug 01 2025

			0.18384376406702461207534175664658126707821355787059...

Ovidiu Furdui, Multiple Fractional Part Integrals and Euler's Constant, Miskolc Mathematical Notes, Vol. 17, No. 1 (2016), pp. 255-266.

Cf. A002117, A002162, A002391, A153810, A386733, A386736, A386737.

RealDigits[9*Log[3]/2 - 6*Log[2] - Zeta[3]/2, 10, 120][[1]]

PARI
```
9*log(3)/2 - 6*log(2) - zeta(3)/2
```

Equals 9*log(3)/2 - 6*log(2) - zeta(3)/2.

A016653 Decimal expansion of log(30).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.401197381662155375413236691606889912248592046451522427768022234605066....

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. A016458 (continued fraction).

RealDigits[Log[30], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(30); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016653.txt", n, " ", d)); \\ Harry J. Smith, May 20 2009

Equals A002391 + A002392. - R. J. Mathar, Jul 22 2025

A016662 Decimal expansion of log(39).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.663561646129646427448732678487844309452758502582956568153739844300958....

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. A016467 (continued fraction).

Cf. A002391, A016636.

Digits:=100: evalf(log(39)); # Wesley Ivan Hurt, Jan 27 2017

RealDigits[Log[39], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(39); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016662.txt", n, " ", d)); \\ Harry J. Smith, May 21 2009

Equals A002391 + A016636. - Michel Marcus, Jan 27 2017

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

A016671 Decimal expansion of log(48).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.871201010907890929064173722755231976949491095263770468217414371611068....

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. A016476 (continued fraction).

RealDigits[Log[48], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(48); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016671.txt", n, " ", d)); \\ Harry J. Smith, May 21 2009

Equals 4*A002162 + A002391. - R. J. Mathar, Jun 07 2024

A016674 Decimal expansion of log(51).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.931825632724325771644779854795652240235693570408494239031932071519786....

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. A016479 (continued fraction).

RealDigits[Log[51],10,120][[1]] (* Harvey P. Dale, Sep 30 2012 *)

default(realprecision, 20080); x=log(51); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016674.txt", n, " ", d)); \\ Harry J. Smith, May 21 2009

Equals A002391 + A016640. - R. J. Mathar, Jun 07 2024

A016677 Decimal expansion of log(54).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			3.988984046564274383602967832225753682017971807828503609324763010405876....

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. A016482 (continued fraction).

RealDigits[Log[54], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(54); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016677.txt", n, " ", d)); \\ Harry J. Smith, May 22 2009

Equals A002162 +3*A002391. - R. J. Mathar, Jun 07 2024

A016680 Decimal expansion of log(57).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			4.04305126783455015140427266881037924188486981912187827027265457004....

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. A016485 (continued fraction).

Equals A002391 + A016642.

evalf(log(57), 100); # Wesley Ivan Hurt, Jan 28 2017

RealDigits[Log[57], 10, 120][[1]] (* Vincenzo Librandi, Jun 21 2015 *)

default(realprecision, 20080); x=log(57); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016680.txt", n, " ", d)); \\ Harry J. Smith, May 22 2009

Equals A002391 + A016642. - R. J. Mathar, Jul 22 2025

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

A121935 Decimal expansion of 1/log(3).

Original entry on oeis.org

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

Offset: 0

Joost de Winter, Sep 03 2006

			0.9102392266268373936142401657361...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers.

Cf. A002391 (log(3)).

SetDefaultRealField(RealField(100)); 1/Log(3); // G. C. Greubel, Oct 07 2019

RealDigits[1/Log[3], 10, 105][[1]] (* Alonso del Arte, Dec 01 2012 *)

d=105;default(realprecision,d+1);print(k=1/log(3));k=10*k;for(c=0,d,z=floor(k);print1(z,",");k=10*(k-z)) \\ Klaus Brockhaus, Sep 06 2006

numerical_approx(1/log(3), digits=100) # G. C. Greubel, Oct 07 2019

Equals 1/log(3).

Equals (1/2) * Product_{k>=1} ((1 + 3^(1/2^k))/2). - Amiram Eldar, Jun 04 2023

More terms from Klaus Brockhaus, Sep 06 2006

cp's OEIS Frontend

A386734 Decimal expansion of Integral_{x=0..1} Integral_{y=0..1} Integral_{z=0..1} {1/(x+y+z)} dx dy dz, where {} denotes fractional part.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A016653 Decimal expansion of log(30).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A016662 Decimal expansion of log(39).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A016668 Decimal expansion of log(45).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A016671 Decimal expansion of log(48).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A016674 Decimal expansion of log(51).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A016677 Decimal expansion of log(54).

Views

Author

Keywords

Examples

References

Links