A381055 -id:A381055 - cp's OEIS frontend

A383131 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; a(n) = -1 if 1 is never reached.

0, 3, 12, 2, 5, 5, 8, 5, 11, 11, 50, 14, 14, 14, 53, 4, 17, 17, 43, 7, 7, 7, 20, 7, 46, 46, 59, 10, 10, 10, 23, 7, 49, 49, 62, 13, 13, 13, 13, 13, 26, 26, 39, 52, 52, 52, 65, 16, 16, 16, 78, 16, 16, 16, 29, 16, 42, 42, 55, 55, 55, 55, 68, 6, 19, 19, 19, 19, 19

Offset: 1

Ya-Ping Lu, Apr 17 2025

nonn

A plot of a(n) for non-zero integers n in the range of (-10000, 10000) is given in LINKS.

Conjecture: a(n) != -1 for any positive or negative integer n.

			a(3) = 12 because it takes 12 steps for n = 3 to reach 1: 3 -> 10 -> -5 -> -14 -> 7 -> 22 -> -11 -> -32 -> 16 -> -8 -> 4 -> -2 -> 1.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Ya-Ping Lu, A plot of a(n) for n in the range of (-10000, 10000)

Cf. A006577, A381055.

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..69);  # Alois P. Heinz, Aug 13 2025

js[n_] := If[EvenQ[n],-n/2,3n+1]; f[n_] := Length[ NestWhileList[js, n, # != 1 &]] - 1; Table[ f[n], {n, 69}] (* James C. McMahon, Apr 30 2025 *)

a(n) = { for (k = 0, oo, if (n==1, return (k), n = if (n%2, 3*n+1, -n/2));); } \\ Rémy Sigrist, Apr 19 2025

def A383131(n):
    ct = 0
    while n != 1: n = 3*n+1 if n%2 else -n>>1; ct += 1
    return ct

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.

Original entry on oeis.org

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

Ryosuke Miyazawa, Aug 06 2025

nonn

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.

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

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

Cf. A006577, A381055, A383131.

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

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 *)

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

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.

cp's OEIS Frontend

A383131 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; a(n) = -1 if 1 is never reached.

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