A134191 - OEIS

%I #7 Mar 09 2016 03:22:29

%S 2,4,5,8,10,11,13,14,16,17,20,22,23,26,28,29,31,32,34,35,38,40,41,44,

%T 46,47,49,50,52,53,56,58,59,61,62,64,65,67,68,70,71,74,76,77,80,82,83,

%U 85,86,88,89,91,92,94,95,98,100,101,103,104,106,107,110,112,113,116,118

%N Impure numbers in the Collatz (3x+1) iteration.

%C Let f(k) be the trajectory of the Collatz iteration of the number k. Then Shaw calls a number n impure if n is in f(k) for some k < n. Shaw has an algorithm for finding congruences that the impure numbers satisfy.

%H T. D. Noe, <a href="/A134191/b134191.txt">Table of n, a(n) for n=1..10000</a>

%H Douglas J. Shaw, <a href="http://www.fq.math.ca/Papers1/44-3/quartshaw03_2006.pdf">The Pure Numbers Generated by the Collatz Sequence</a>, The Fibonacci Quarterly, Vol. 44, Number 3, August 2006, pp. 194-201.

%F Complement of A061641.

%e The Collatz trajectory of 3 is (3,10,5,16,8,4,2,1), showing that the numbers 4,5,8,10,16 are impure.

%t c[n_] := If[EvenQ[n], n/2, 3n + 1]; nn=1000; t=Table[0,{nn}]; Do[If[t[[n]]==0, m=n; While[m=c[m]; If[nn>=m>n && t[[m]]==0, t[[m]]=n]; m>nn || t[[m]]>0]], {n,nn}]; Flatten[Position[t,_?(#>0&)]]

%K nonn

%O 1,1

%A _T. D. Noe_, Oct 12 2007