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A344276 Number of halving and tripling steps to reach 3 in the '3x+3' problem, or -1 if 3 is never reached.

%I A344276 #33 Jun 13 2021 06:05:28
%S A344276 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 A344276 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 A344276 20,20,7,20,20,20,7,108,7,7,8,28,15,15,23,15,23,23,10
%N A344276 Number of halving and tripling steps to reach 3 in the '3x+3' problem, or -1 if 3 is never reached.
%C A344276 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 A344276 From _Jon E. Schoenfield_, May 14 2021: (Start)
%C A344276 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 A344276 .
%C A344276     m   (Sum_{n=1..m} a(n))/m  (Sum_{n=1..3m} c(n))/3m
%C A344276   ----  ---------------------  -----------------------
%C A344276   10^1       6.7000000000            8.6666666667
%C A344276   10^2      31.4200000000           32.1466666667
%C A344276   10^3      59.5420000000           58.9020000000
%C A344276   10^4      84.9666000000           84.6180333333
%C A344276   10^5     107.5384000000          107.6915966667
%C A344276 where c(n) = A006577(n) is the number of steps in the '3x+1' trajectory of n.
%C A344276 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 A344276 Rémy Sigrist, <a href="/A344276/b344276.txt">Table of n, a(n) for n = 1..10000</a>
%F A344276 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 A344276 a(1) = 2, with the trajectory 1 -> 6 -> 3.
%e A344276 a(5) = 9, with the trajectory 5 -> 18 -> 9 -> 30 -> 15 -> 48 -> 24 -> 12 -> 6 -> 3.
%p A344276 a:= proc(n) a(n):= 1+a(`if`(n::odd, 3*n+3, n/2)) end: a(3):=0:
%p A344276 seq(a(n), n=1..100);  # _Alois P. Heinz_, May 14 2021
%t A344276 If[#!=3,#0@If[OddQ@#,3#+3,#/2]+1,0]&/@Range@100 (* _Giorgos Kalogeropoulos_, May 14 2021 *)
%o A344276 (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 A344276 Cf. A067896 (trajectory of 41).
%Y A344276 Cf. A006577 (3x+1 steps).
%K A344276 nonn
%O A344276 1,1
%A A344276 _Matthias van Sligtenhorst_, May 13 2021