login
This site is supported by donations to The OEIS Foundation.

 

Logo

Many excellent designs for a new banner were submitted. We will use the best of them in rotation.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002391 Decimal expansion of natural logarithm of 3.
(Formerly M4595 N1960)
17
1, 0, 9, 8, 6, 1, 2, 2, 8, 8, 6, 6, 8, 1, 0, 9, 6, 9, 1, 3, 9, 5, 2, 4, 5, 2, 3, 6, 9, 2, 2, 5, 2, 5, 7, 0, 4, 6, 4, 7, 4, 9, 0, 5, 5, 7, 8, 2, 2, 7, 4, 9, 4, 5, 1, 7, 3, 4, 6, 9, 4, 3, 3, 3, 6, 3, 7, 4, 9, 4, 2, 9, 3, 2, 1, 8, 6, 0, 8, 9, 6, 6, 8, 7, 3, 6, 1, 5, 7, 5, 4, 8, 1, 3, 7, 3, 2, 0, 8, 8, 7, 8, 7, 9, 7 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,3

REFERENCES

W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. 2.

Uhler, Horace S.; Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17. Proc. Nat. Acad. Sci. U. S. A. 26, (1940). 205-212.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Harry J. Smith, Table of n, a(n) for n = 1..20000

Eric Weisstein's World of Mathematics, BBP-Type Formula

D. H. Bailey, Compendium to BBP formulas

G. Huvent, Formules BBP en base 3 [From Jaume Oliver Lafont, Oct 12 2009]

_Simon Plouffe_, Plouffe's Inverter, The natural logarithm of 3 to 10000 digits

Simon Plouffe, log(3), natural logarithm of 3 to 2000 places

S. Ramanujan, Notebook entry

FORMULA

log(3) = sum_{n>=1} (9*n-4)/((3*n-2)*(3*n-1)*3*n). [Jolley, Summation of Series, Dover (1961) eq 74]

ln(3) = 1/4*(1+ Sum((1/(9)^(k+1))*(27/(2*k+1) + 4/(2*k+2) + 1/(2*k+3)), k = 0 .. infinity) ) (a BBP-type formula) [From Alexander R. Povolotsky, Dec 01 2008]

log(3) = 4/5 +2/10*sum((1/4)^n*(1/(2*n+1)+1/(2*n+3)),n=0...infinity) [From Alexander R. Povolotsky, Dec 18 2008]

log(3)=sum((1/9)^(k+1)(9/(2k+1)+1/(2k+2)),k=0..infinity) [From Jaume Oliver Lafont, Dec 22 2008]

Sum_{i>=1} 1/(9^i*i) + Sum_{i>=0} 1/(9^i*(i+1/2)) = 2*log(3) (Huvent 2001) [From Jaume Oliver Lafont, Oct 12 2009]

log(3) = sum(k>=1, A191907(3,k)/k ). (conjecture) [From Mats Granvik, Jun 19 2011]

EXAMPLE

1.098612288668109691395245236922525704647490557822749451734694333637494...

MATHEMATICA

RealDigits[Log[3], 10, 120][[1]]  (* Harvey P. Dale, Apr 23 2011 *)

PROG

(PARI) log(3) \\ Charles R Greathouse IV, Jan 24 2012

CROSSREFS

Cf. A058962, A154920, A002162, A016731 (continued fraction).

Sequence in context: A059068 A059069 A084660 * A193626 A087044 A105415

Adjacent sequences:  A002388 A002389 A002390 * A002392 A002393 A002394

KEYWORD

nonn,cons,changed

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Editing and more terms from Charles R Greathouse IV, Apr 20 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified April 23 04:11 EDT 2014. Contains 240909 sequences.