A213198 Number of iterations of the map n -> f(f(f(...f(n)...))) to reach the end of the cycle, where f(n) = A006577(n), the initial number n is not counted.

0, 1, 5, 2, 0, 7, 4, 6, 7, 8, 11, 8, 8, 10, 10, 3, 9, 6, 6, 5, 5, 11, 11, 9, 12, 9, 13, 7, 7, 7, 10, 1, 10, 9, 9, 6, 6, 6, 10, 7, 11, 7, 8, 4, 4, 4, 10, 12, 10, 10, 10, 12, 12, 7, 7, 7, 2, 7, 2, 7, 7, 15, 15, 8, 14, 14, 14, 11, 11, 11, 14, 12, 12, 12, 11, 12

Offset

1,3

Comments

A006577 is the number of halving and tripling steps to reach 1 in '3x+1' problem.

The end of the cycle is 1 or 5 for n = 5, 32, 57, 59, 344, 346, 348, 349, ...

Links

Michel Lagneau, Table of n, a(n) for n = 1..10000

Example

a(3) = 5 because the 5 iterations to reach 1 are A006577(3) = 7; A006577(7) = 16; A006577(16) = 4; A006577(4) = 2; A006577(2) = 1.
a(5) = 0 because A006577(5) = 5 is the end of the cycle.
a(57) = 2 because A006577(57) = 32 and A006577(32) = 5 is the end of the cycle.

Maple

for n from 1 to 200 do:
    m:=n: a:=2:
    for it from 1 to 1000
    while (a>1) do:
        jj:=0: a:=0: x:=m:
        if m=5 then
            printf(`%d, `, it-1): jj:=1:
        else
            for i from 1 to 1000
            while (x>1) do:
                if irem(x, 2)=0 then
                    x := x/2: a := a+1:
                else
                    x := 3*x+1: a := a+1:
                fi:
            od:
            m:=a:
        fi:
    od:
    if jj=0 then
        printf(`%d, `, it-1):
    fi:
od:

Mathematica

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; f[n_] := Length[Collatz[n]] - 1; Table[k = Rest[NestWhileList[f, n, UnsameQ, All]]; If[k[[1]] == n, 0, k = DeleteCases[k, 0]; If[Length[k] > 1 && k[[-1]] == k[[-2]], k = Most[k]]; Length[k]], {n, 100}] (* T. D. Noe, Mar 01 2013 *)

Crossrefs

Cf. A006577.

Sequence in context: A147710 A153456 A021872 * A021196 A257406 A269980

Adjacent sequences: A213195 A213196 A213197 * A213199 A213200 A213201

Keyword

nonn

Author

Michel Lagneau, Mar 01 2013

Status

approved