A224166 -id:A224166 - cp's OEIS frontend

A224399 Numbers k such that A224166(k) = A008908(k).

1, 2, 4, 8, 13, 16, 26, 32, 35, 52, 64, 70, 81, 93, 104, 128, 140, 162, 181, 186, 208, 241, 256, 280, 324, 362, 372, 416, 455, 482, 483, 512, 543, 560, 607, 643, 645, 648, 724, 744, 809, 815, 832, 903, 910, 914, 915, 964, 966, 967, 1024, 1079, 1081, 1086, 1087

Offset: 1

Michel Lagneau, Apr 05 2013

nonn

Numbers k for which the number of iterations starting with -k to reach the last number of the cycle equals the number of iterations starting with k to reach 1 in Collatz (3x+1) trajectory of +/-k.

			a(6) = 26 because A224166(26) = A008908(26) = 11.
The trajectory of - 26 is :
-26 -> - 13 -> -38 -> -19 -> -56 -> -28 -> -14 -> -7 -> -20 -> -10 -> -5 with 11 iterations (the first value is counted);
The trajectory of 26 is :
26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 with 11 iterations (the first value is counted).

Cf. A008908, A224166.

z:={1}:
  for m from -1 by -1 to -1500 do:
   lst:={m}:a:=0: x:=m: lst:=lst union {x}:
        for i from 1 to 100 do:
         lst:=lst union {x}:
         if irem(abs(x), 2)=1
         then
         x:=3*x+1: lst:=lst union {x}:
         else
         x:=x/2: lst:=lst union {x}:
         fi:
         od:
         n0:=nops(lst):
         if lst intersect z = {1}
         then
         n1:=n0-2:
         else
         n1:=n0-1:
      fi:
       a:=0:y:=-m:
          for it from 1 to 100 while (y>1) do:
             if irem(y,2)=0
             then
             y := y/2:a:=a+1:
             else
              y := 3*y+1: a := a+1:
             fi:
          od:
          if n1=a
          then
          printf(`%d, `,-m):
          else
          fi:
od:

A224254 Full cycle lengths in the Collatz (3x+1) problem when the negative integers are used.

Original entry on oeis.org

2, 2, 2, 2, 5, 2, 5, 2, 5, 5, 2, 2, 5, 5, 2, 2, 18, 5, 5, 5, 18, 2, 18, 2, 18, 5, 5, 5, 2, 2, 18, 2, 18, 18, 5, 5, 18, 5, 2, 5, 18, 18, 2, 2, 18, 18, 5, 2, 18, 18, 5, 5, 2, 5, 18, 5, 2, 2, 2, 2, 18, 18, 5, 2, 2, 18, 18, 18, 2, 5, 2, 5, 18, 18, 5, 5, 2, 2, 2, 5, 5

Offset: 1

Michel Lagneau, Apr 02 2013

nonn

There are other cycles of lengths 2, 5 and 18 if negative integers are used. In Z, it is conjectured that the five values of cycle are 1, 2, 3, 5 and 18 (see A121510).

			a(1) = 2 because the cycle -1 -> -2 -> -1... contains 2 distinct terms;
a(5) = 5 because the cycle -5 -> -14 -> -7->-20 -> -5 ... contains 5 distinct terms;
a(17) = 18 because the cycle -17 -> -50 -> -25->-74 -> -37 -> -110 -> -55->-164 -> -82 -> -41 -> -122->-61 -> -182 -> -91 -> -272->-136 -> -68 -> -34 -> -17... contains 18 distinct terms.

Wikipedia, Collatz conjecture

Cf. A121510, A224166, A224183.

import sympy
def A224254(n):
  return next(sympy.cycle_length(lambda x:3*x+1 if x%2 else x//2,-n))[0] # Pontus von Brömssen, Jan 24 2021

Data corrected by Pontus von Brömssen, Jan 24 2021

A346369 a(n) is the number of steps the Collatz trajectory of -n takes to reach a cycle, or -1 if no cycle is ever reached.

Original entry on oeis.org

0, 0, 3, 1, 0, 4, 0, 2, 7, 0, 5, 5, 5, 0, 8, 3, 0, 8, 3, 0, 6, 6, 1, 6, 0, 6, 3, 1, 9, 9, 4, 4, 11, 0, 9, 9, 0, 4, 12, 1, 0, 7, 7, 7, 5, 2, 12, 7, 9, 0, 7, 7, 15, 4, 0, 2, 28, 10, 10, 10, 0, 5, 15, 5, 36, 12, 3, 0, 18, 10, 18, 10, 7, 0, 5, 5, 13, 13, 13, 2, 18

Offset: 1

Felix Fröhlich, Jul 14 2021

sign

The analog of A139399 for negative starting values.

Is a(n) > -1 for all n?

			For n = 5: The trajectory of -5 starts -5, -14, -7, -20, -10, -5, -14, -7, -20, -10, -5, ..., with -5 already being part of the cycle {-5, -14, -7, -20, -10}, so a(5) = 0.
For n = 6: The trajectory of -6 starts -6, -3, -8, -4, -2, -1, -2, -1, -2, -1, -2, ..., reaching a term of the cycle {-2, -1} after 4 steps, so a(6) = 4.

Cf. A139399, A224166, A224183, A224254.

a006370(n) = if(n%2==0, n/2, 3*n+1)
trajectory(n, terms) = my(v=[n]); while(#v < terms, v=concat(v, a006370(v[#v]))); v
a(n) = my(t, v=[]); for(k=1, oo, t=trajectory(-n, k); for(x=1, #t, if(x < #t && t[x]==t[#t], return(x-1))))

a(n) = A224166(n) - A224254(n) = 1 + A224183(n) - A224254(n). - Pontus von Brömssen, Jul 24 2021

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A224399 Numbers k such that A224166(k) = A008908(k).

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A224254 Full cycle lengths in the Collatz (3x+1) problem when the negative integers are used.

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A346369 a(n) is the number of steps the Collatz trajectory of -n takes to reach a cycle, or -1 if no cycle is ever reached.

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