What is notable about the composite numbers between twin primes? Look at the composites between twin primes (A014574):
$$
4, 6, 12, 18, 30, 42, 60, 72, 102, 108, 138, 150, 180, 192, 198, 228, \\ 240, 270, 282, 312, 348, 420, 432, 462, 522, 570, 600, 618, 
\ldots \;.
$$
Is there anything special about their distribution of factors,
number of divisors, or other number-theoretical properties?
Or are these twin-prime averages totally "normal" numbers, as far as we know?
 A: Yes there is something interesting about the composite number between twin primes. Let $p,p+2$ be a twin prime pair. Then $p-1$ is the composite number just preceding the pair, $p+1$ is the composite number between the pair and $p+3$ is the composite number just after the pair. Let $d(n)$ and $\omega(n)$ be the number of divisors and the number of distinct prime factors of $n$ respectively. Then, experimental data shows that 
$$
2.27 \sum_{p,p+2 \in P}{d(p-1)} \approx 2.27 \sum_{p,p+2 \in P}{d(p+3)} \approx \sum_{p,p+2 \in P}{d(p+1)}
$$
$$
1.24 \sum_{p,p+2 \in P}{\omega(p-1)} \approx 1.24 \sum_{p,p+2 \in P}{\omega(p+3)} \approx \sum_{p,p+2 \in P}{\omega(p+1)}
$$
i.e. roughly speaking, the composite number between a twin prime pair has on an average $24\%$ more distinct prime factors than the composite number just before or just after the pair and more than twice as many divisors.


*

*Why does the middle composite have significantly more divisors and prime factors than its composite neighbors?

*Why does the just composite before a twin prime have roughly the same number of divisors or prime factors as the composite just after the twin prime?


I shall post this in a separate question with the detailed data not only for twin prime but similar observations for other prime gaps.
A: Other than $n=4$, $n$ has the property that for every prime number $q=6k\pm 1$, $\frac{n}{6}\not \equiv \pm k \bmod q$.
Let $m:=\frac{n}{6};\ 6m=n$ and the twin primes $(n-1),(n+1)$ are represented as $(6m-1),(6m+1)$
We see from the properties of the semiprime $(6m-1)(6m+1)=36m^2-1$. If $36m^2-1$ is not a semiprime, it is divisible by some prime $6k\pm 1,\ k\ne m$. In other words $36m^2-1=(6k\pm 1)(6j\pm 1)$ where $k,j\ne m$. Expanding, $36m^2-1=36jk\pm 6j \pm 6k-1$ which reduces to $6m^2=6jk\pm j \pm k$
This means $m^2=jk \pm \frac{j\pm k}{6}$, and since $m^2$ is an integer, $\frac{j\pm k}{6}$ must be an integer, call it $r$. So $j=6r\pm k$ and we can substitute this back to obtain $m^2=(6r\pm k)k \pm r=k^2\pm r(6k\pm 1)$.
Finally we obtain $m^2-k^2=(m-k)(m+k)=r(6k\pm 1)$. This means that $(m-k)(m+k)\equiv 0 \bmod q$ which can only be true if $m\equiv \pm k \bmod q$.
So if $36m^2-1$ is not a semiprime, then $m\equiv \pm k \bmod q$ when $q\mid 36m^2-1$. But if $(6m-1),(6m+1)\in \mathbb P$ then $36m^2-1$ is a semiprime, so for every prime number $q=6k\pm 1$, $m\not \equiv \pm k \bmod q \Rightarrow \frac{n}{6}\not \equiv k \bmod q$.
A: $n-1,n+1$ are both primes iff for all prime $p\le \sqrt{n}, n \not \equiv \pm 1\bmod p$.
The random model for the primes, from which we predict the density of twin primes, is that those conditions are more or less independent from one $p$ to the other, ie. $n$ has nothing else special.
A: My conjecture is that there is a „local sieve“ around a twin prime pair such that all prime factors up to the square root of the higher prime are to find in the surrounding composite numbers in a maximum distance equal to the highest prime below this square root, relative to the number between the twin prime pair. If that is guaranteed, there are simply no prime factors left, and the pair must be prime. Since the number between the pair must be divisible by 6, it is always at least one to two primes ahead of its surrounding composites, and it is allowed to accumulate a lot more prime factors since neighboring integers cannot have common divisors, which also lets other composites in the surroundings accumulate prime factors. Does this, in average, give a Gaussian? - Of course, that‘s just an idea and a very coarse outline of a „local sieve“ for twin primes or primes in general.
