login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

a(n) = floor(a(n-1)/2) if this is not among 0, a(1), ..., a(n-2); otherwise a(n) = 3*a(n-1).
30

%I #32 Sep 17 2023 21:00:14

%S 1,3,9,4,2,6,18,54,27,13,39,19,57,28,14,7,21,10,5,15,45,22,11,33,16,8,

%T 24,12,36,108,324,162,81,40,20,60,30,90,270,135,67,201,100,50,25,75,

%U 37,111,55,165,82,41,123,61,183,91,273,136,68

%N a(n) = floor(a(n-1)/2) if this is not among 0, a(1), ..., a(n-2); otherwise a(n) = 3*a(n-1).

%C This permutation of the natural numbers is the multiply-and-divide (MD) sequence for (M,D)=(3,2). The "MD question" is this: for relatively prime M and D, does the MD sequence contain every positive integer exactly once? An affirmative proof for the more general condition that log base D of M is irrational is given by Mateusz Kwaśnicki in Crux Mathematicorum 30 (2004) 235-239. - _Clark Kimberling_, Jun 30 2004

%H T. D. Noe, <a href="/A050000/b050000.txt">Table of n, a(n) for n = 1..10000</a>

%H Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/unsolved.html">Unsolved Problems and Rewards</a>.

%H Mateusz Kwaśnicki, <a href="http://www.im.pwr.wroc.pl/~kwasnicki/stuff/MDproblem.en.pdf">The solution of M-D problem</a> (2008).

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%t a[0] = 0; a[1] = 1; a[n_] := a[n] = (b = Floor[a[n-1]/2]; If[FreeQ[Table[ a[k], {k, 0, n-2}], b], b, 3*a[n-1]]);

%t Array[a, 60] (* _Jean-François Alcover_, Jul 13 2016 *)

%o (Haskell)

%o a050000 n = a050000_list !! (n-1)

%o a050000_list = 1 : f [1,0] where

%o f xs'@(x:xs) | x `div` 2 `elem` xs = 3 * x : f (3 * x : xs')

%o | otherwise = x `div` 2 : f (x `div` 2 : xs')

%o -- _Reinhard Zumkeller_, Nov 13 2011

%Y Cf. A050076, A050001 (inverse).

%Y MD sequences:

%Y A050076 (2,3), A050124 (2,5),

%Y this sequence (3,2), A050104 (3,4),

%Y A050080 (4,3),

%Y A050004 (5,2), A050084 (5,3), A050108 (5,4),

%Y A050008 (6,2), A050088 (6,3), A050112 (6,4),

%Y A050012 (7,2), A050092 (7,3),

%Y A050096 (8,3),

%Y A050016 (9,2),

%Y A050020 (10,2), A050100 (10,3).

%K nonn,nice,easy

%O 1,2

%A _Clark Kimberling_