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

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

Author

N. J. A. Sloane

Keywords

• nonn
• base
• easy
• nice

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.

Crossrefs

First differences give A151910.

Cf. A055642, A000244.

Programs

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

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 *)

Extensions

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