A386885 a(n) is the number of iterations that -n requires to reach 1 under the map x -> -x/2 if x is even, 3x + 1 if x is odd.

2, 1, 4, 4, 10, 13, 13, 3, 16, 6, 6, 6, 45, 9, 9, 6, 48, 12, 12, 12, 25, 51, 51, 15, 15, 15, 15, 15, 41, 54, 54, 5, 18, 18, 18, 18, 31, 44, 44, 8, 57, 8, 8, 8, 21, 21, 21, 8, 34, 47, 47, 47, 60, 60, 60, 11, 11, 11, 11, 11, 86, 24, 24, 8, 37, 50, 50, 50, 50, 63

Offset

1,1

Comments

Equivalently, a(n) is the number of iterations that n requires to reach -1 under the map x -> -x/2 if x is even, 3x - 1 if x is odd. This is called the shadow Collatz map in the Ryosuke Miyazawa reference.

Conjecture: Every nonzero integer reaches 1. Verified numerically up to +-5*10^7.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10001

Ryosuke Miyazawa, The Shadow Collatz Conjecture: The Journey of All Integers Toward -1, preprint, Zenodo, (2025).

Formula

a(1) = 2; for n > 1, a(n) = 1 + a(f(n)), where f(n) = -n/2 if n is even, 3n - 1 if n is odd.

Example

n = 7 requires a(7) = 13 steps to reach -1: 7 -> 20 -> -10 -> 5 -> 14 -> -7 -> -22 -> 11 -> 32 -> -16 -> 8 -> -4 -> 2 -> -1.

Maple

a:= proc(n) option remember; `if`(n=-1, 0,
      1+a(`if`(n::even, -n/2, 3*n-1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 13 2025

Mathematica

a[n_]:=Module[{s = 0, m = n}, While[m!=-1, s++; m=If[BitAnd[m, 1]==1, 3*m -1, -m/2]]; s]; Array[a, 70] (* James C. McMahon, Aug 18 2025 *)

Prog

(PARI)  a(n) = if(n==0, oo, my(s=0); while(n!=-1, s++; n=if(bitand(n, 1), 3*n-1, -n/2)); s) \\ Andrew Howroyd, Aug 06 2025

Crossrefs

Cf. A006577, A381055, A383131.

Sequence in context: A241513 A095293 A034409 * A209478 A211178 A048049

Adjacent sequences: A386882 A386883 A386884 * A386886 A386887 A386888

Keyword

nonn

Author

Ryosuke Miyazawa, Aug 06 2025

Status

approved