A002392 -id:A002392 - cp's OEIS frontend

A016653 Decimal expansion of log(30).

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

A016723 Decimal expansion of log(100).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			4.605170185988091368035982909368728415202202977257545952066655801935145....

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. A016528 (continued fraction).

SetDefaultRealField(RealField(100)); 2*Log(10); // G. C. Greubel, Sep 15 2018

RealDigits[2*Log[10], 10,100][[1]] (* G. C. Greubel, Sep 15 2018 *)

default(realprecision, 20080); x=log(100); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016723.txt", n, " ", d)); \\ Harry J. Smith, May 25 2009

Equals 2*A002392.

A059546 Beatty sequence for log(10)/(log(10)-1).

Original entry on oeis.org

1, 3, 5, 7, 8, 10, 12, 14, 15, 17, 19, 21, 22, 24, 26, 28, 30, 31, 33, 35, 37, 38, 40, 42, 44, 45, 47, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 67, 68, 70, 72, 74, 76, 77, 79, 81, 83, 84, 86, 88, 90, 91, 93, 95, 97, 98, 100, 102, 104, 106, 107, 109, 111, 113, 114, 116

Offset: 1

Mitch Harris, Jan 22 2001

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 A059545.

Cf. A002392.

Floor[Range[100]*(1 + 1/(Log[10] - 1))] (* Paolo Xausa, Jul 05 2024 *)

{ default(realprecision, 100); b=log(10)/(log(10) - 1); for (n = 1, 2000, write("b059546.txt", n, " ", floor(n*b)); ) } \\ Harry J. Smith, Jun 28 2009

a(n) = floor(n*log(10)/(log(10) - 1)). - Michel Marcus, Jan 04 2015

A062542 Decimal expansion of the continued fraction constant (base 10).

Original entry on oeis.org

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

Offset: 1

Jason Earls, Jun 25 2001

"(By strange coincidence, the information in a typical continued fraction term is very nearly one decimal digit - actually pi^2/(6 (ln 2) (ln 10)) = 1.0306.) R. W. Gosper. Math-Fun list, Apr 9 1998. This constant is the average number of decimal digits necessary to have the equivalent continued fraction representations of a number in base 10. In other words if you have N decimal digits it will give you N/C = N/1.0306 valid partial quotients in average." - Simon Plouffe

			1.03064083410071293588177609411693684092592031112...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 1.8 Khintchine-Lévy constants, p. 60.

Simon Plouffe, Plouffe's Inverter.
Eric Weisstein's World of Mathematics, Lochs' Theorem.

Cf. A062543.

Cf. A002162, A002392, A013661.

RealDigits[Pi^2/(6Log[2]Log[10]),10,120][[1]] (* Harvey P. Dale, Apr 11 2012 *)

Pi^2/(6*log(2)*log(10)) \\ Stefano Spezia, Nov 16 2024

Equals Pi^2/(6 (log 2) (log 10)).

Equals A013661/(A002162*A002392). - Stefano Spezia, Nov 16 2024

A228240 Integer lengths of log(10)-primes.

Original entry on oeis.org

1, 2, 40, 242, 842, 1541, 75067

Offset: 1

Eric W. Weisstein, Aug 17 2013

Next term > 76902. - Eric W. Weisstein, Oct 12 2015

			log(10) = 2.302585093..., so
a(1) = 1 (1-digit number 2 is prime),
a(2) = 2 (2-digit number 23 is a prime),
a(3) = 40 (40-digit number 2302585092994045684017991454684364207601 is prime).

Eric Weisstein's World of Mathematics, Constant Primes
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Natural Logarithm of 10 Digits

Cf. A228241 (log(10)-primes).

Cf. A002392 (decimal digits of log(10)).

Module[{nn=2000,lg10},lg10=RealDigits[Log[10],10,nn][[1]];IntegerLength/@ Select[Table[FromDigits[Take[lg10,n]],{n,nn}],PrimeQ]] (* Harvey P. Dale, Jul 16 2015 *)

a(n) = IntegerLength(A228241(n)).

a(7) from Eric W. Weisstein, Oct 12 2015

A228241 Log 10-primes: primes in the initial decimal digits of log(10).

Original entry on oeis.org

2, 23, 2302585092994045684017991454684364207601

Offset: 1

Eric W. Weisstein, Aug 17 2013

The next term (a(4)) has 242 digits. - Harvey P. Dale, Sep 13 2021

Eric W. Weisstein, Table of n, a(n) for n = 1..5
Eric Weisstein's World of Mathematics, Constant Primes
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Natural Logarithm of 10 Digits

Cf. A228240 (integer lengths of log(10)-primes).

Cf. A002392 (decimal expansion of log(10)).

Module[{nn=500,l10},l10=RealDigits[Log[10],10,nn][[1]];Select[ Table[ FromDigits[ Take[l10,n]],{n,nn}],PrimeQ]] (* Harvey P. Dale, Sep 13 2021 *)

A059182 Engel expansion of log(10) = 2.30259...

Original entry on oeis.org

1, 1, 4, 5, 20, 30, 48, 74, 265, 818, 897, 2027, 5107, 6846, 13049, 236586, 364437, 643493, 1144424, 7294777, 49484843, 51161394, 76008087, 202870914, 391981014, 11731070977, 79069971960, 415100034571, 1212266245583

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.
.

G. C. Greubel, Table of n, a(n) for n = 1..1000
F. Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
P. Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
Index entries for sequences related to Engel expansions

Cf. A002392.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]]; EngelExp[N[Log[10], 7!], 50] (* modified by G. C. Greubel, Dec 26 2016 *)

A111769 Decimal expansion of abs(log_10(tan 1 degree)).

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Nov 21 2005

			1.75807853138637225488535519037064202097844798645087074642159375066
598330424067054925599054790998656775347490626861719117

Cf. A019810; A019898.

RealDigits[Abs[Log[10,Tan[1 Degree]]],10,120][[1]] (* Harvey P. Dale, Aug 21 2016 *)

Equals A111767/A002392 - R. J. Mathar, Apr 23 2009

Clarified base of logarithm in definition - R. J. Mathar, Apr 23 2009

A117028 Decimal expansion of abs(log_10(sine of 1 radian)).

Original entry on oeis.org

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

Offset: 0

Mohammad K. Azarian, Apr 15 2006

			0.07496085456049550613838945425940469659487454046535873354155675712422

Cf. A049469; A073447.

Equals A117029/A002392. - R. J. Mathar, Apr 23 2009

Clarified base of logarithm in definition - R. J. Mathar, Apr 23 2009

A197071 Decimal expansion of Pi/log(10).

Original entry on oeis.org

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

Offset: 1

Philippe Deléham, Oct 09 2011

Pi/log(10) = 1.364376353841841...

Cf. A000796, A002392.

Mathematica
```
RealDigits[Pi/Log[10], 10, 101][[1]]
```

a(52) corrected and more terms from Georg Fischer, Apr 04 2020

cp's OEIS Frontend

A016653 Decimal expansion of log(30).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A016723 Decimal expansion of log(100).

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A059546 Beatty sequence for log(10)/(log(10)-1).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A062542 Decimal expansion of the continued fraction constant (base 10).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A228240 Integer lengths of log(10)-primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A228241 Log 10-primes: primes in the initial decimal digits of log(10).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

A059182 Engel expansion of log(10) = 2.30259...

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica