*This note is related to*


[Can all three numbers $\ n\ \ n^2-1\ \ n^2+1\ $ be fine (as opposed to coarse)?](https://mathoverflow.net/questions/396310/can-all-three-numbers-n-n2-1-n21-be-fine-as-opposed-to-coarse)




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Let
$$ m\ n\ \in\ \mathbb N\_{\_{>1}}\ :=\ \{x\in\mathbb Z: x>1\} $$
be arbitrary.   Let $\ P(n)\ $ be the largest prime divisor of $n$.


**Definition:**   Molecularity of $n$ is
$$ M(n)\ :=\ \log\_{P(n)}(n) $$


Instantly,


**Theorem**


* $ M(n) \ge 1;$
* $ M(n)=1\quad\Leftrightarrow\quad p\ $ is a prime;
* $ M(n^k)\ =\ k\cdot M(n)\qquad $ (for every $\ k=1\ 2\ \ldots);$
* $ M(m\cdot n)\ \le\ M(m)+M(n);$
* $ M(m\cdot n)=M(m)+M(n)\quad\Leftrightarrow\quad
 P(\gcd(m\ n))\ =\ P(m\cdot n). $


For instance:
$$ n>3\quad\Rightarrow\quad M(n^2-1)\ <\ M(n-1)+M(n+1) $$


**Question 1:**   What is
$$ \sup\_{n>2}\ \min(M(n-1)\,\ M(n)\,\ M(n+1))\quad ?$$
**Question 2:**   What is
$$ \inf\_{n>2}\ \frac1{M(n-1)}+\frac1{M(n)}+\frac1{M(n+1)}
 \quad? $$


The ever-sharper bounds would be greatly appreciated.


I dare, this time with a greater probability, that the above sup is $\ \le 4,\ $ and that $4$ cannot be actually attained.



Molecularity $\ M(n)$
For every $u>0$, there exists $n$ such that each of $P(n-1)$, $P(n)$, $P(n-1)$ is less than $n^u$. This was proved by Eggleton and Selfridge (Consecutive integers with no large prime factors, J. Austral. Math. Soc. Ser. A 22 (1976), 1–11). In fact their proof is constructive (see pp. 2-3 of their paper). It follows that the supremum in Question 1 is infinite, while the infimum in Question 2 is zero.


I should add that this phenomenon also holds for an arbitrary long string of consecutive integers. For example, there exists $n$ such that each of $P(n-50)$, $P(n-49)$, ..., $P(n+50)$ is less than $n^u$. This was proved by Balog and Wooley (On strings of consecutive integers with no large prime factors, J. Austral. Math. Soc. Ser. A 64 (1998), 266–276).


