A127885 a(n) = minimal number of steps to get from n to 1, where a step is x -> 3x+1 if x is odd, or x -> either x/2 or 3x+1 if x is even; or -1 if 1 is never reached.

0, 1, 7, 2, 5, 8, 16, 3, 11, 6, 14, 9, 9, 17, 17, 4, 12, 12, 20, 7, 7, 15, 15, 10, 23, 10, 23, 10, 18, 18, 31, 5, 18, 13, 13, 13, 13, 21, 26, 8, 21, 8, 21, 16, 16, 16, 29, 11, 16, 16, 24, 11, 11, 24, 24, 11, 24, 19, 24, 19, 19, 32, 32, 6, 19, 19, 27, 14, 14, 14, 27, 14, 27, 14, 14, 22, 22, 27, 27, 9, 22, 22, 22, 9, 9, 22, 22, 17, 22, 17, 30, 17, 17, 30, 30, 12, 30, 17, 17, 17

Offset

1,3

Comments

In contrast to the "3x+1" problem (see A006577), here you are free to choose either step if x is even.

See A125731 for the number of steps in the reverse direction, from 1 to n.

References

M. J. Halm, Sequences (Re)discovered, Mpossibilities 81 (Aug. 2002), p. 1.

Links

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

Index entries for sequences related to 3x+1 (or Collatz) problem

Formula

a(1) = 0; and for n > 1, if n is odd, a(n) = 1 + a(3n+1), and if n is even, a(n) = 1 + min(a(3n+1),a(n/2)). [But with a similar caveat as in A257265] - Antti Karttunen, Aug 20 2017

Example

Several early values use the path:
6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1.
The first path where choosing 3x+1 for even x helps is:
9 -> 28 -> 85 -> 256 -> 128 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1.

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 A127885(n, lim) == lim, all you know is that the value is >= lim). To use it, do manual iteration on the limit.
A127885 := proc(n, lim) local d, d2; options remember;
if (n = 1) then return 0; end if;
if (lim <= 0) then return 0; end if;
if (n > 2 ^ lim) then return lim; end if;
if (n mod 2 = 0) then
d := A127885(n/2, lim-1);
d2 := A127885(3*n+1, d);
if (d2 < d) then d := d2; end if;
else
d := A127885(3*n+1, lim-1);
end if;
return 1+d;
end proc;

Mathematica

Table[-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 *)

Prog

(PARI) { A127885(n) = my(S, k); S=[n]; k=0; while( S[1]!=1, k++; S=vecsort( concat(apply(x->3*x+1, S), apply(x->x\2, select(x->x%2==0, S) )), , 8);  ); k } /* Max Alekseyev, Sep 03 2015 */

Crossrefs

A127886 gives the difference between A006577 and this sequence.

Cf. A006577, A125731, A127887, A125195, A125686, A125719, A261870.

Cf. A290100 (size of the final set when using Alekseyev's algorithm).

Cf. also A257265.

Sequence in context: A072761 A337357 A340420 * A006577 A337150 A280234

Adjacent sequences: A127882 A127883 A127884 * A127886 A127887 A127888

Keyword

nonn

Author

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

Extensions

Escape clause added to definition by N. J. A. Sloane, Aug 20 2017

Status

approved