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 A260389 For the Collatz (3x+1) iteration starting at k, the sequence lists the smallest k such that Fibonacci(n) belongs to the trajectory of k, or 0 if no such k exists. 1

%I

%S 2,2,3,6,3,3,7,42,7,73,39,288,27,27,135,1974,1419,861,2787,13530,7297,

%T 5247,33963,92736,100033,161857,116395

%N For the Collatz (3x+1) iteration starting at k, the sequence lists the smallest k such that Fibonacci(n) belongs to the trajectory of k, or 0 if no such k exists.

%C The initial term k is not counted as an element of the trajectory.

%e a(1)=a(2)=2 because 1 = Fibonacci(1)=Fibonacci(2) is in the trajectory 2 -> 1;

%e a(3)=3 because 2 = Fibonacci(3) is in the trajectory 3 -> 10 -> 5 -> ... -> 2 ->1;

%e a(4)=6 because 7 = Fibonacci(4)=3 is in the trajectory 6 -> 3 -> 10 -> ... -> 1;

%e a(5)=3 because 5 = Fibonacci(5)=3 is in the trajectory 3 -> 10 -> 5 -> ... -> 1.

%e a(6)=3 because 8 = Fibonacci(6)=3 is in the trajectory 3 -> 10 -> 5 -> 16 -> 8 -> ... -> 1;

%e a(7)=7 because 13 = Fibonacci(7)=3 is in the trajectory 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> ... -> 1.

%p with(numtheory):with(combinat,fibonacci):

%p for n from 1 to 10000 do:

%p jj:=0:

%p for k from 2 to 10^8 while(jj=0) do:

%p lst:={}:m:=k:ii:=0:it:=10^6:

%p for i from 1 to it while(ii=0) do:

%p if irem(m,2)=0

%p then

%p m:=m/2:

%p else m:=3*m+1:

%p fi:

%p lst:=lst union {m}:

%p if m=1

%p then

%p ii:=1:

%p else

%p fi:

%p od:

%p n0:=nops(lst):

%p for j from 1 to n0 while(jj=0)do:

%p if fibonacci(n)=lst[j]

%p then

%p jj:=1:

%p else fi:

%p od:

%p if jj=1 then printf("%d %d \n",n,k):

%p else fi:

%p od:od:

%Y Cf. A000045, A006577.

%K nonn,hard

%O 1,1

%A _Michel Lagneau_, Nov 22 2015

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Last modified June 21 19:59 EDT 2021. Contains 345365 sequences. (Running on oeis4.)