This question came up while looking at the well known lower bound for $\phi(n)$, the Euler Totient function
$$ \phi(n) \gt {n \over {e^\gamma \ln \ln n} + {3 \over \ln \ln n}} \tag{1} $$
For $m$ such that $gcd(m, n) = 1$, we have
$$ \phi(mn) = \phi(m) \phi(n) \tag{2} $$
If we take $m = p_k\#$, the $k$-th primorial and $\gcd(p_k\#, n) = 1$, we have
$$ \phi(p_k\#) \phi(n) \gt {{p_k\# \times n} \over {e^\gamma \ln \ln (p_k\# \times n)} + {3 \over \ln \ln (p_k\# \times n)}} \tag{3} $$
Therefore,
$$ \phi(n) \gt {p_k\# \over \phi(p_k\#)} \times {n \over {e^\gamma \ln \ln (p_k\# \times n)} + {3 \over \ln \ln (p_k\# \times n)}} \tag{4} $$
I tried $n = 61997851 = 7919 \times 7829$ as an example. $\phi(n) = 7918 \times 7828 = 61982104$ using the standard formula.
The known lower bound formula gives us $\phi(n) \gt 3578918$ which is quite far off. I then estimate $\phi(n)$ using the various primorial values. The table of the estimated $\phi(n)$ for various values of $k$ is given below:
| $k$ for $k$-th primorial | $\phi(n)$ estimate |
|---|---|
| 5 | 44494923 |
| 50 | 59431412 |
| 100 | 60063451 |
| 110 | 60106645 |
| 150 | 60279934 |
| 195 | 60398410 |
| 197 | 60402559 |
| 199 | 60406310 |
| 200 | 60408296 |
I used WolframAlpha to do the calculations. Here is the link to an example for $p_{110}\#$ on WolframAlpha.
The question I have is: Is there a way we can ensure that the lower bounds for $\phi(n)$ evaluate to a sharper bound upon successive iterations using the above method or combining with an upper bound formula? (such as $\phi(n) \le n - \sqrt{n}$ for composite $n$).
For instance, in the same example, if we used the upper bound formula, we get:
$$\phi(n) \le 61989978$$
Can we then find the range for $k$ that satisfies the inequality and then search for the factor of $n$ within that range?
$$ n - \sqrt{n} \ge \phi(n) \gt {p_k\# \over \phi(p_k\#)} \times {n \over {e^\gamma \ln \ln (p_k\# \times n)} + {3 \over \ln \ln (p_k\# \times n)}} \tag{5} $$
EDIT: An earlier version of this question had an incorrect computation of $\phi(n)$ which spawned an inquiry into a different issue. The current question is focused on approaches for improving the bounds with other iterative techniques.