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Number of halving and tripling steps to reach 3 in the '3x+3' problem, or -1 if 3 is never reached.
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%I #33 Jun 13 2021 06:05:28

%S 2,3,0,4,9,1,4,5,7,10,10,2,18,5,5,6,21,8,8,11,16,11,11,3,11,19,19,6,

%T 19,6,6,7,14,22,22,9,22,9,9,12,9,17,17,12,17,12,12,4,25,12,12,20,113,

%U 20,20,7,20,20,20,7,108,7,7,8,28,15,15,23,15,23,23,10

%N Number of halving and tripling steps to reach 3 in the '3x+3' problem, or -1 if 3 is never reached.

%C The '3x+3' problem is a slight variation of the Collatz problem. If n is even, divide it by 2, if n is odd, multiply by 3 and add 3. The number of steps to reach 3 are given, which may be the endpoint for all n (empirical observation).

%C From _Jon E. Schoenfield_, May 14 2021: (Start)

%C It seems that the average number of steps among the '3x+3' trajectories for n in 1..3m is close to the average number of steps in the '3x+1' trajectories for n in 1..m:

%C .

%C m (Sum_{n=1..m} a(n))/m (Sum_{n=1..3m} c(n))/3m

%C ---- --------------------- -----------------------

%C 10^1 6.7000000000 8.6666666667

%C 10^2 31.4200000000 32.1466666667

%C 10^3 59.5420000000 58.9020000000

%C 10^4 84.9666000000 84.6180333333

%C 10^5 107.5384000000 107.6915966667

%C where c(n) = A006577(n) is the number of steps in the '3x+1' trajectory of n.

%C Perhaps a good way to explain this result is that, other than the values connected by the string of consecutive divide-by-2 steps at the beginning of the trajectory of an even number not divisible by 3, every value in every '3x+3' trajectory is a multiple of 3, so within any given interval, there are only about 1/3 as many values available for inclusion in '3x+3' trajectories as there are in '3x+1' trajectories. (End)

%H Rémy Sigrist, <a href="/A344276/b344276.txt">Table of n, a(n) for n = 1..10000</a>

%F a(3) = 0; for all other n > 0, if n is even, a(n) = a(n/2) + 1; if n is odd, a(n) = a(3n+3) + 1.

%e a(1) = 2, with the trajectory 1 -> 6 -> 3.

%e a(5) = 9, with the trajectory 5 -> 18 -> 9 -> 30 -> 15 -> 48 -> 24 -> 12 -> 6 -> 3.

%p a:= proc(n) a(n):= 1+a(`if`(n::odd, 3*n+3, n/2)) end: a(3):=0:

%p seq(a(n), n=1..100); # _Alois P. Heinz_, May 14 2021

%t If[#!=3,#0@If[OddQ@#,3#+3,#/2]+1,0]&/@Range@100 (* _Giorgos Kalogeropoulos_, May 14 2021 *)

%o (PARI) a(n) = for (k=0, oo, if (n==3, return (k), n%2==0, n=n/2, n=3*n+3)) \\ _Rémy Sigrist_, Jun 13 2021

%Y Cf. A067896 (trajectory of 41).

%Y Cf. A006577 (3x+1 steps).

%K nonn

%O 1,1

%A _Matthias van Sligtenhorst_, May 13 2021