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

Author

Michel Lagneau, Mar 01 2013

Keywords

• nonn

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, ...

Examples

			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.
		

Links

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

Crossrefs

Cf. A006577.

Programs

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 *)