The total stopping time
σ
∞
(
n
)
\sigma _{\infty }(n)
of a positive integer
n
n
is the minimal number of iterates of the
3
x
+
1
3x+1
function needed to reach the value
1
1
, and is
+
∞
+\infty
if no iterate of
n
n
reaches
1
1
. It is shown that there are infinitely many positive integers
n
n
having a finite total stopping time
σ
∞
(
n
)
\sigma _{\infty }(n)
such that
σ
∞
(
n
)
>
6.14316
log
n
.
\sigma _{\infty }(n) > 6.14316 \log n.
The proof involves a search of
3
x
+
1
3x +1
trees to depth 60, A heuristic argument suggests that for any constant
γ
>
γ
B
P
≈
41.677647
\gamma > \gamma _{BP} \approx 41.677647
, a search of all
3
x
+
1
3x +1
trees to sufficient depth could produce a proof that there are infinitely many
n
n
such that
σ
∞
(
n
)
>
γ
log
n
.
\sigma _{\infty }(n)>\gamma \log n.
It would require a very large computation to search
3
x
+
1
3x + 1
trees to a sufficient depth to produce a proof that the expected behavior of a “random”
3
x
+
1
3x +1
iterate, which is
γ
=
2
log
4
/
3
≈
6.95212
,
\gamma =\frac {2}{\log 4/3} \approx 6.95212,
occurs infinitely often.