A125731 a(n) = minimal number of steps to get from 1 to n, where a step is x -> 3x+1 if x is odd, or x -> either x/2 or 3x+1 if x is even. Set a(n) = -1 if n cannot be reached from 1.

0, 2, -1, 1, 6, -1, 3, 8, -1, 5, 5, -1, 2, 15, -1, 7, 7, -1, 12, 4, -1, 4, 9, -1, 9, 9, -1, 14, 14, -1, 6, 19, -1, 6, 11, -1, 11, 11, -1, 3, 29, -1, 16, 16, -1, 8, 8, -1, 8, 21, -1, 8, 13, -1, 26, 13, -1, 13, 13, -1, 5, 31, -1, 18, 18, -1, 5, 23, -1, 10

Offset

1,2

Comments

In contrast to the "3x+1" problem, here you are free to choose either step if x is even.

Clearly a(3k) = -1 for all k; we conjecture that a(n) >= 0 otherwise.

See A127885 for the number of steps in the reverse direction, from n to 1.

Links

David Applegate, Table of n, a(n) for n = 1..1000

Example

The initial values use these paths:
1 -> 4 -> 2 -> 7 -> 22 -> 11.
1 -> 4 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8.
1 -> 4 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 49 -> 148 -> 74 -> 37 -> 12 -> 56 -> 28 -> 14.

Maple

# Code from David Applegate: Be careful - the function takes an iteration limit and returns the limit if it wasn't able to determine the answer (that is, if A125731(n, lim) == lim, all you know is that the value is >= lim). To use it, do manual iteration on the limit.
A125731 := proc(n, lim) local d, d2; options remember;
if (n = 1) then return 0; end if;
if (n mod 3 = 0) then return -1; end if;
if (lim <= 0) then return 0; end if;
if (n > (3 ** (lim+1) - 1)/2) then return lim; end if;
if (n mod 9 = 4 or n mod 9 = 7) then
d := A125731((n-1)/3, lim-1);
d2 := A125731(2*n, d);
if (d2 < d) then d := d2; end if;
else
d := A125731(2*n, lim-1);
end if;
return 1+d;
end proc;

Crossrefs

Sequence in context: A053383 A181538 A322128 * A123361 A265315 A179380

Adjacent sequences: A125728 A125729 A125730 * A125732 A125733 A125734

Keyword

sign

Author

David Applegate and N. J. A. Sloane, Feb 02 2007

Status

approved