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%I #11 Nov 08 2017 12:27:03
%S 1,3,3,8,3,6,9,17,4,20,7,15,10,10,18,18,5,13,21,21,8,8,16,16,11,24,11,
%T 112,19,19,19,107,6,27,14,14,22,22,22,35,9,110,9,30,17,17,17,105,12,
%U 25,25,25,12,12,113,113,20,33,20,33,20,20,108,108,7,28,28
%N Length (= size) of the orbit of n under the "3x+1" map A006370: x -> x/2 if even, 3x+1 if odd. a(n) = -1 in case the orbit would be infinite.
%C The orbit of x under f is O(x; f) = { f^k(x); k = 0, 1, 2, ... }, i.e., the set of all points in the trajectory of x under iterations of f.
%C The famous "3x+1 problem" or Collatz conjecture (also attributed to other names) states that for f = A006370, the trajectory (f^k(x); k >= 0) always ends in the cycle 1 -> 4 -> 2 -> 1, for any integer starting value x >= 0.
%e a(0) = 1 = # { 0 }, since 0 -> 0 -> 0 ... under A006370.
%e a(1) = 3 = # { 1, 4, 2 }, since 1 -> (3*1 + 1 =) 4 -> 2 -> 1 -> 4 etc. under A006370.
%e a(3) = 8 = # { 3, 10, 5, 16, 8, 4, 2, 1 }, since 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 -> 4 etc. under A006370.
%Y Cf. A006370 (Collatz or 3x+1 map), A008908 (number of steps to reach 1), A174221 (the "PrimeLatz" map: add 3 next primes), A293980, A293975 (variant: add the next prime), A293982.
%K nonn
%O 0,2
%A _M. F. Hasler_, Nov 05 2017
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