A213185 - OEIS

%I #8 Feb 28 2013 16:15:58

%S 0,0,0,0,0,1,2,0,0,1,1,0,0,2,3,0,0,3,1,0,0,1,2,0,0,1,1,0,0,3,4,0,0,2,

%T 1,2,0,1,2,0,0,1,1,1,0,2,3,0,0,2,1,0,0,1,2,0,0,2,1,1,0,4,5,0,0,3,1,1,

%U 0,1,2,0,0,1,1,1,0,2,3,0,0,5,1,0,0,1,2

%N For the Collatz (3x+1) iteration starting at n, number of integers k such that T(n,k) = T(n-1,k) +1 where T(n,k) is the k-th number of the trajectory (n is not counted).

%C T(n,k) is periodic, for example

%C T(6,3) = 10 and T(7,3) = 11;

%C T(10,3) = 16 and T(11,3) = 17,

%C …………..

%C T(6+4m, 3) = 10 + 6m and T(6+4m+1, 3) = 10 + 6m+1.

%C .................

%H Michel Lagneau, <a href="/A213185/b213185.txt">Table of n, a(n) for n = 1..10000</a>

%e a(7) = 2 because the Collatz iterations starting at 7 and 6 are :

%e T(7,k) =[7, 22, 11, 34, 17, 52,  26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]

%e T(6,k) =[6, 3, 10, 5, 16, 8,  4, 2, 1] ; the number 7 is not counted

%e => T(7,3) = T(6,3) + 1 and T(7,5) = T(6,5) + 1, hence a(7) = 2.

%p nn:=200:T:=array(1..nn):U:=array(1..nn): aa:=1:

%p for i from 1 to nn do:

%p    U[i]:=0:

%p od:

%p       for n from 1 to nn do:

%p           a:=0:k:=0:x:=n:

%p                for i from 1 to 10^8 while (x>1) do:

%p                   if irem(x,2)=0 then

%p                   x := x/2:a:=a+1:k:=k+1:T[k]:=x:

%p               else

%p                  x := 3*x+1: a := a+1: k:=k+1:T[k]:=x:

%p                fi:

%p             od:

%p            it:=0:

%p                 for j from 1 to min(a,aa) do:

%p                    if T[j]=U[j]+1 then

%p                    it:=it+1:

%p                  else

%p                  fi:

%p                 od:

%p                      for m from 1 to a do:

%p                         U[m]:=T[m]:

%p                      od:

%p           aa:=a: printf(`%d, `,it): od:

%Y Cf. A006577, A213186.

%K nonn

%O 1,7

%A _Michel Lagneau_, Feb 28 2013