1,9

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.

a(n) = A006577(n) - A127885(n)

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

A127887 gives the indices of the nonzero entries.

Sequence in context: A124272 A128820 A067485 this_sequence A085121 A079204 A036482

Adjacent sequences: A127883 A127884 A127885 this_sequence A127887 A127888 A127889

nonn

David Applegate (david(AT)research.att.com) and N. J. A. Sloane (njas(AT)research.att.com), Feb 04 2007