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%I #16 Aug 20 2017 10:55:12
%S 0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,88,8,0,0,75,0,8,
%T 0,0,8,8,0,8,0,88,0,8,0,0,0,75,0,8,8,0,0,0,88,88,8,8,0,8,0,0,75,75,0,
%U 8,8,0,0,0,0,75,8
%N Steps saved by choice in "3x+1" iteration.
%C Normal "3x+1" iteration requires x->x/2 if x is even. a(n) is the number of iterations that can be saved by also allowing x->3x+1 if x is even.
%H Antti Karttunen, <a href="/A127886/b127886.txt">Table of n, a(n) for n = 1..1000</a> (computed from the b-files of A006577 and A127885)
%F a(n) = A006577(n) - A127885(n).
%e a(9) = 8 because for 9 the traditional 3x+1 iteration follows the 19-step path:
%e 9 -> 28 -> 14 -> 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
%e while allowing x->3x+1 for even x gives the 11-step path:
%e 9 -> 28 -> 85 -> 256 -> 128 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1
%t Table[Length@ NestWhileList[If[OddQ@ #, 3 # + 1, #/2] &, n, # > 1 &] - Length@ NestWhileList[Flatten[# /. {k_ /; OddQ@ k :> 3 k + 1, k_ /; EvenQ@ k :> {k/2, 3 k + 1}}] &, {n}, FreeQ[#, 1] &], {n, 126}] (* _Michael De Vlieger_, Aug 20 2017 *)
%Y Cf. A006577, A127885, A127887 (gives the indices of the nonzero entries).
%K nonn
%O 1,9
%A _David Applegate_ and _N. J. A. Sloane_, Feb 04 2007
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