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%I #11 Aug 11 2014 22:45:23
%S 1,2,5,2,3,6,11,2,12,4,9,6,5,12,11,2,7,12,13,4,3,10,9,6,14,6,74,12,11,
%T 12,71,2,15,8,7,12,13,14,23,4,73,4,17,10,8,10,69,6,14,14,15,6,5,74,73,
%U 12,19,12,21,12,11,72,72,2,15,16,17,8,7,8,67,12,75,14,6,14,13,24,23,4
%N Number of squarefree integers in {n, f(n), f(f(n)), ...., 1}, where f is the Collatz function defined by f(x) = x/2 if x is even; f(x) = 3x + 1 if x is odd.
%C Number of squarefree terms in 3x+1 trajectory started at n.
%H Harvey P. Dale, <a href="/A078372/b078372.txt">Table of n, a(n) for n = 1..1000</a>
%H J. C. Lagarias, <a href="http://www.cecm.sfu.ca/organics/papers/lagarias/paper/html/paper.html">The 3x+1 problem and its generalizations</a>, Amer. Math. Monthly, 92 (1985), 3-23.
%e The finite sequence n, f(n), f(f(n)), ...., 1 for n = 12 is 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, which has six squarefree terms. Hence a(12) = 6.
%e n=61: trajectory={61,184,92,46,23,70,35,...,20,10,5,16,8,4,2,1}, squarefree terms={61,46,23,70,35,106,53,10,5,2,1}, so a(61)=11.
%t Table[Count[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&], _?(SquareFreeQ)], {n,80}] (* _Harvey P. Dale_, Oct 23 2011 *)
%K nonn
%O 1,2
%A _Joseph L. Pe_, Dec 24 2002
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