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A067433 Triangle in which row n gives trajectory of n under the map k -> k/3 if k is divisible by 3, otherwise k -> next multiple of 3, stopping when reaching 1 (the initial term n is not included).

%I A067433 #18 Dec 23 2021 10:04:12
%S A067433 1,3,1,1,6,2,3,1,6,2,3,1,2,3,1,9,3,1,9,3,1,3,1,12,4,6,2,3,1,12,4,6,2,
%T A067433 3,1,4,6,2,3,1,15,5,6,2,3,1,15,5,6,2,3,1,5,6,2,3,1,18,6,2,3,1,18,6,2,
%U A067433 3,1,6,2,3,1,21,7,9,3,1,21,7,9,3,1,7,9,3
%N A067433 Triangle in which row n gives trajectory of n under the map k -> k/3 if k is divisible by 3, otherwise k -> next multiple of 3, stopping when reaching 1 (the initial term n is not included).
%C A067433 These numbers converge to various last 3-digit endings and only to 2 last 2-digit numbers: 2,1 or 3,1. m=3. p=1 below. If m=2, p=1 you get the x+1 conjecture. If m=2, p=3 you get the 3x+1 conjecture. See Link for numbers with a large number of digits. Other conjectures are possible by trial-and-error input of m and n. It is interesting to note that for many m and p=m+1 the program converges to 1. However, for m prime and p=m+1 the program always converges to m^2, m, 1. Also for m+1 prime the program converges to m^2, m, 1 most of the time. An exception is m=6. The sequence converges but to what I call an uninteresting ending.
%H A067433 Cino Hilliard, <a href="http://groups.msn.com/BC2LCC/page.msnw?fc_p=%2Fkx%2Bp%20problems&amp;fc_a=0">The x+1 conjecture</a> [Dead link]
%e A067433 4 -> 6 -> 2 -> 3 -> 1, so row 4 is 6,2,3,1. Row 5 is the same.
%t A067433 nxt[n_]:=If[Divisible[n,3],n/3,3(Floor[n/3]+1)]; Join[{1},Flatten[ Table[ Rest[ NestWhileList[nxt,i,#!=1&]],{i,30}]]] (* _Harvey P. Dale_, Sep 16 2012 *)
%o A067433 (PARI) multxp2(n,m,p) = { print1(1" "); for(j=1,n, x=j; c=0; while(x>1, r = x%m; if(r==0,x=x/m,x=x*p+m-r); print1(x" "); ); ) }
%Y A067433 Cf. A080816.
%K A067433 easy,tabf,nonn
%O A067433 1,2
%A A067433 _Cino Hilliard_, Mar 29 2003
%E A067433 Corrected by _Harvey P. Dale_, Sep 16 2012