A001682 Numbers k such that 3^k, 3^(k+1) and 3^(k+2) have the same number of digits. (Formerly M5109 N2213)

0, 21, 42, 65, 86, 109, 130, 151, 174, 195, 218, 239, 262, 283, 304, 327, 348, 371, 392, 415, 436, 457, 480, 501, 524, 545, 568, 589, 610, 633, 654, 677, 698, 721, 742, 763, 786, 807, 830, 851, 874, 895, 916, 939, 960, 983, 1004, 1027, 1048

Offset

1,2

Comments

Equivalently the fractional part of n*log(3) lies between 0 and 1 - 2*log(3), about 0.04576; 1 - 2*log(3) is also the density of the sequence. - Kevin Costello, Aug 08 2002

References

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Murray Klamkin and Joe Lipman, Problem E1238, Amer. Math. Monthly, 64 (1957), 367.

Mathematica

Select[Range[0, 2000], IntegerLength[3^#] == IntegerLength[3^(#+1)] == IntegerLength[3^(#+2)]&] (* Jean-François Alcover, Nov 24 2011 *)
Flatten[Position[Partition[IntegerLength[3^Range[0, 1100]], 3, 1], _?( Length[ Union[#]]==1&), {1}, Heads->False]]-1 (* Harvey P. Dale, Jan 31 2015 *)
SequencePosition[IntegerLength[3^Range[0, 1200]], {x_, x_, x_}][[All, 1]]-1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 12 2018 *)

Prog

(Haskell)
a001682 n = a001682_list !! (n-1)
a001682_list = [k | k <- [0..], let m = 3^k, a055642 m == a055642 (9*m)]
-- Reinhard Zumkeller, Oct 10 2011

Crossrefs

First differences give A151910.

Cf. A055642, A000244.

Sequence in context: A008603 A235497 A086794 * A180963 A355406 A078440

Adjacent sequences: A001679 A001680 A001681 * A001683 A001684 A001685

Keyword

nonn,base,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from R. K. Guy and Emeric Deutsch, Mar 09 2005

Status

approved