OFFSET
1,9
COMMENTS
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.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..1000 (computed from the b-files of A006577 and A127885)
EXAMPLE
a(9) = 8 because for 9 the traditional 3x+1 iteration follows the 19-step path:
9 -> 28 -> 14 -> 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1
while allowing x->3x+1 for even x gives the 11-step path:
9 -> 28 -> 85 -> 256 -> 128 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
David Applegate and N. J. A. Sloane, Feb 04 2007
STATUS
approved