

A127886


Steps saved by choice in "3x+1" iteration.


5



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, 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, 8, 8, 0, 0, 0, 0, 75, 8
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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 bfiles of A006577 and A127885)


FORMULA

a(n) = A006577(n)  A127885(n).


EXAMPLE

a(9) = 8 because for 9 the traditional 3x+1 iteration follows the 19step 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 11step 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

Cf. A006577, A127885, A127887 (gives the indices of the nonzero entries).
Sequence in context: A128820 A067485 A180225 * A270033 A085121 A228634
Adjacent sequences: A127883 A127884 A127885 * A127887 A127888 A127889


KEYWORD

nonn


AUTHOR

David Applegate and N. J. A. Sloane, Feb 04 2007


STATUS

approved



