Perhaps you like the following overview.
I'll write for a number $a_1$ and its smallest ancestor $a_2$, which is larger than or equal to $a_1$ and is also not divisible by $3$.
This can then be thought to be iterated. For instance, beginning at $a_1=5$, iterating $2$ times gives the following protocol:
values: exponents at 2 along the iteration
a1 a3 : A1 A2
5 17 : 3 2
that means $ 5 \to (5 \cdot 2^3-1)/3=13 \to (13 \cdot 2^2 -1 )/3 = 17 $
Here a protocol of the first $27$ examples of $a_1=6 k -1$ :
a1 a33 | A1 A2 A3 ... Exponents at 2 ... A32
-------------------------+-------------------------------------------------------- --------------------------------------+
5 1629567600864557 | 3 2 5 2 4 4 2 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 |
11 1847830689651265 | 3 3 3 4 2 5 4 2 3 3 4 2 2 3 3 3 2 5 4 2 5 2 3 2 3 3 3 3 4 4 2 |
17 5794018136407313 | 5 2 4 4 2 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3 3 |
23 30467312081069 | 3 4 2 2 2 2 5 4 4 2 3 3 2 3 5 2 3 2 3 2 4 2 3 2 3 3 2 2 5 2 2 |
29 9855097011473413 | 3 3 4 2 5 4 2 3 3 4 2 2 3 3 3 2 5 4 2 5 2 3 2 3 3 3 3 4 4 2 2 |
35 23896770660498613 | 5 2 3 3 3 4 4 4 4 4 4 2 5 2 3 3 4 2 2 2 4 2 2 2 3 2 2 3 4 4 2 |
41 868065190823725 | 3 2 2 2 3 2 2 3 3 2 5 2 3 3 2 4 2 5 2 5 2 5 2 4 4 4 4 2 2 4 2 |
47 8011680485691313 | 3 5 2 2 3 5 4 2 3 3 5 2 2 5 4 2 2 2 3 3 2 4 4 2 3 3 2 2 3 5 4 |
53 4528745657817329 | 5 4 4 2 3 2 2 2 3 5 2 3 3 3 3 2 3 5 2 2 4 2 2 5 4 2 3 4 2 2 5 |
59 5022658183850245 | 3 2 3 5 2 2 2 3 2 4 2 2 3 3 4 4 2 4 2 4 4 4 2 3 4 2 2 4 4 4 2 |
65 1385166667016593 | 3 3 3 3 2 2 3 5 2 5 4 2 4 4 4 2 3 3 2 4 2 3 3 2 4 2 2 3 4 2 3 |
71 757921508018869 | 5 2 2 2 3 3 3 2 3 4 4 4 2 3 3 5 4 2 2 2 3 3 2 5 2 2 2 4 2 2 2 |
77 13140129348631217 | 3 4 2 5 4 2 3 3 4 2 2 3 3 3 2 5 4 2 5 2 3 2 3 3 3 3 4 4 2 2 4 |
83 1769460185153089 | 3 3 2 3 3 2 4 2 3 5 4 2 3 4 2 5 2 4 2 2 5 2 4 2 3 3 3 3 2 4 2 |
89 15209936237556805 | 5 2 3 4 4 2 2 3 3 2 2 3 2 5 2 3 2 2 4 4 4 4 2 3 5 2 2 5 2 3 3 |
95 1012199105165357 | 3 2 2 5 2 2 5 2 3 2 3 5 2 4 4 4 4 2 3 4 2 2 2 3 3 3 3 2 3 3 2 |
101 4312339992160045 | 3 5 4 2 4 2 3 3 2 5 2 2 3 3 4 2 5 2 2 3 3 3 4 4 2 2 3 3 2 4 2 |
107 146334932561525941 | 5 4 2 2 5 2 2 3 3 4 2 3 5 2 3 3 2 3 4 2 3 4 4 2 3 3 3 3 4 4 2 |
113 38559608325447409 | 3 2 3 4 2 3 2 4 4 2 4 4 2 2 3 2 5 2 3 3 3 5 2 5 2 2 5 4 2 3 5 |
119 10160472862670533 | 3 3 5 2 3 3 4 4 2 5 2 2 4 2 2 2 2 4 2 4 4 4 2 2 2 3 2 3 2 5 4 |
125 10682240647588417 | 5 2 2 3 5 4 2 3 3 5 2 2 5 4 2 2 2 3 3 2 4 4 2 3 3 2 2 3 5 4 2 |
131 89511465278846773 | 3 4 4 4 2 5 4 2 2 3 3 2 2 5 2 4 4 2 2 3 4 2 5 2 2 2 3 3 5 2 3 |
137 2922724885389493 | 3 3 2 2 2 2 2 3 5 2 2 4 4 2 2 4 2 5 2 4 2 4 4 4 2 5 2 2 3 3 2 |
143 97785619677512965 | 5 2 5 2 3 4 2 3 3 3 3 2 2 2 4 2 3 5 2 5 2 4 2 3 2 5 2 5 2 5 2 |
149 1589973825711857 | 3 2 4 2 5 2 3 3 4 2 3 3 3 5 2 3 3 2 3 3 2 3 3 3 2 4 2 2 3 3 5 |
155 6620575296987905 | 3 5 2 3 2 2 2 3 4 2 2 3 2 2 5 2 5 2 5 2 4 4 4 2 4 4 2 4 4 2 2 |
- - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - +
and here the same for $a_1 = 6 k +1$
a1 a33 | A1 A2 A3 ... Exponents at 2 ... A32
-------------------------+-------------------------------------------------------- --------------------------------------+
7 292183593823813 | 4 2 2 3 3 3 3 2 2 3 5 2 5 4 2 4 4 4 2 3 3 2 4 2 3 3 2 4 2 2 3 |
13 4345513602305485 | 2 5 2 4 4 2 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3 |
19 399563157372085 | 2 4 4 4 2 2 4 4 2 5 2 2 4 2 2 3 2 3 3 3 3 2 3 3 2 2 3 3 4 4 2 |
25 532750876496113 | 4 4 4 2 2 4 4 2 5 2 2 4 2 2 3 2 3 3 3 3 2 3 3 2 2 3 3 4 4 2 5 |
31 325524446558897 | 2 3 2 2 2 3 2 2 3 3 2 5 2 3 3 2 4 2 5 2 5 2 5 2 4 4 4 4 2 2 4 |
37 389578125098417 | 2 2 3 3 3 3 2 2 3 5 2 5 4 2 4 4 4 2 3 3 2 4 2 3 3 2 4 2 2 3 4 |
43 14667849204846277 | 4 2 5 2 5 2 2 5 4 2 2 3 5 4 2 2 2 2 3 2 4 2 3 2 2 3 4 2 5 4 4 |
49 1038875000262445 | 2 3 3 3 3 2 2 3 5 2 5 4 2 4 4 4 2 3 3 2 4 2 3 3 2 4 2 2 3 4 2 |
55 72788213540101 | 2 2 4 2 3 2 2 4 4 4 2 5 2 3 2 2 3 2 3 2 2 2 5 2 3 4 2 2 3 5 2 |
61 81246165549517 | 4 2 2 2 2 5 4 4 2 3 3 2 3 5 2 3 2 3 2 4 2 3 2 3 3 2 2 5 2 2 3 |
67 2851863044541901 | 2 5 2 3 4 4 2 2 3 3 2 2 3 2 5 2 3 2 2 4 4 4 4 2 3 5 2 2 5 2 3 |
73 97050951386801 | 2 4 2 3 2 2 4 4 4 2 5 2 3 2 2 3 2 3 2 2 2 5 2 3 4 2 2 3 5 2 4 |
79 863744967943647473 | 4 4 2 3 4 2 5 4 4 2 4 2 2 2 5 2 2 5 2 3 5 4 2 3 4 4 2 3 5 2 5 |
85 28919706244085557 | 2 3 2 3 4 2 3 2 4 4 2 4 4 2 2 3 2 5 2 3 3 3 5 2 5 2 2 5 4 2 3 |
91 967757600546545 | 2 2 5 4 2 3 2 3 5 2 3 4 2 3 5 4 2 3 2 4 4 2 3 3 2 2 2 2 2 3 5 |
97 1035210148125877 | 4 2 3 2 2 4 4 4 2 5 2 3 2 2 3 2 3 2 2 2 5 2 3 4 2 2 3 5 2 4 2 |
103 274005458005265 | 2 3 3 2 2 2 2 2 3 5 2 2 4 4 2 2 4 2 5 2 4 2 4 4 4 2 5 2 2 3 3 |
109 4629681017726533 | 2 2 2 3 2 2 3 3 2 5 2 3 3 2 4 2 5 2 5 2 5 2 4 4 4 4 2 2 4 2 3 |
115 613915116385969 | 4 2 4 2 2 3 4 4 2 3 3 3 2 3 2 2 3 2 2 3 5 2 4 4 2 3 2 4 4 2 4 |
121 1290343467395393 | 2 5 4 2 3 2 3 5 2 3 4 2 3 5 4 2 3 2 4 4 2 3 3 2 2 2 2 2 3 5 2 |
127 173264499591143213 | 2 4 2 2 5 2 5 2 3 2 4 2 5 2 3 2 4 4 2 5 2 3 3 3 4 4 2 5 4 4 2 |
133 710334501994817 | 4 4 2 2 4 4 2 5 2 2 4 2 2 3 2 3 3 3 3 2 3 3 2 2 3 3 4 4 2 5 2 |
139 11852812255905349 | 2 3 4 2 2 3 3 2 4 4 2 3 2 2 4 4 4 2 3 4 2 3 4 4 2 5 2 2 5 2 3 |
145 24691632094541509 | 2 2 3 2 2 3 3 2 5 2 3 3 2 4 2 5 2 5 2 5 2 4 4 4 4 2 2 4 2 3 4 |
151 25802620180311985 | 4 2 3 5 4 2 2 2 5 2 2 2 4 4 4 2 5 4 2 3 2 2 2 4 2 3 5 2 2 5 4 |
157 6696877578466993 | 2 3 5 2 2 2 3 2 4 2 2 3 3 4 4 2 4 2 4 4 4 2 3 4 2 2 4 4 4 2 4 |
- - + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - +
Notes (just some scribbled thoughts, q&d):
Of course, the vectors of exponents have unbounded length.
Even if $a_1$ is member of a nontrivial cycle, the vector of exponents is not periodic because it cannot contain decreasing subsequences of $a_k$ (by design of the routine)
Most of the $a_1$ shown on some row in the protocol occur as $a_k$ in an earlier row of the protocol, so the exponents-vectors are usually simply trailing parts of vectors of earlier rows.
- But not all: odd numbers $a_1$ which are result of $(3 a_2+1)/2$ are not in the trailing part of earlier $a_1$ , but as well have infinite exponents-vectors.
This answers also the question whether all $a_1 $ not divisible by $3$ have infinitely (iterated) ancestors.
It might be fun to detect patterns in the $k$'th columns of exponents $A_k$. Of course $A_1$ and $A_2$ are simple periodics, but I didn't look at this deeper.
My idea of a Pari/GP-script is
{nextexpo(a0,it=1)=my(a1=a0,a2,A,vA); vA=vector(it);
for(k=1,it,
if(a1 % 3 ==1, a2=(4*a1-1)/3);
if(a1 % 3 ==2, a2=(2*a1-1)/3;if(a2<a1,a2=4*a2+1)); \\make sure a2 is >= a1!
if(a2 % 3==0,a2=4*a2+1); \\ if a3 divisible by 3, exponent must be increased by 2
A = valuation(3*a2+1,2);
vA[k]=A; a1=a2;
);
return(concat([a0,a2],vA));}
\\ now generate protocol
forstep(a1=7,165,6,print(nextexpo(a1,32)))
Added
A protocol of the subsequent $a_k$ beginning at $a_1=5$ shows how the later exponents-vectors are trailing vectors of the earlier ones:
a1 a33 | A1 A2 A3 ... Exponents at 2 ... A32
-------------------------+-------------------------------------------------------- --------------------------------------+
5 1629567600864557 3 2 5 2 4 4 2 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3
13 4345513602305485 2 5 2 4 4 2 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3
17 5794018136407313 5 2 4 4 2 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3
181 61802860121678005 2 4 4 2 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3
241 329615253982282693 4 4 2 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3
1285 439487005309710257 4 2 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3
6853 1171965347492560685 2 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3
9137 12500963706587313973 3 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3
24365 16667951608783085297 3 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3
64973 44447870956754894125 3 3 3 2 5 2 3 4 2 4 4 4 2 4 2 3 4 2 3 2 5 2 3