Every odd number lies between two even numbers. Accordingly we have two categories of consecutive even number pairs; those pairs which surround primes and those pairs which surround odd composites. Some even numbers can belong to both categories as explained in the example below.
E.g: The pair $(8,10)$ will fall in the category of composite since it contains the odd composite number $9$. The pair $(10, 12)$ belongs to the category of primes since they contain the prime $11$. Hence there will be some overlap on the boundaries of primes as is the case with $10$ in this example. As primes thin out, such overlaps will also thin out accordingly.
Data: Experimental data shows that the even numbers which surround a prime have on a average about $28\%$ more divisors and $7\%$ more distinct prime factors than the even numbers which surround odd composites. For numbers up to $3.5 \times 10^7$,
- The average number of divisors of the even pairs surrounding primes is $35.39$ while that of those which surround odd composite numbers is only $27.70$.
- Moreover, difference between the average number of distinct prime factors of these two categories seems to converge to a value in the neighborhood of $0.27$
Question 1: How or why does the act of surrounding a prime give the two surrounding even numbers a higher number of divisors and distinct prime factors?
n = 3 pa = pb = ca = cb = 0 ip = ic = 0 target = step = 10^6 while true: if is_prime(n) == True: ip = ip + 1 pb = pb + len(divisors(n-1)) pa = pa + len(divisors(n+1)) else: ic = ic + 1 cb = cb + len(divisors(n-1)) ca = ca + len(divisors(n+1)) if n > target: print n, ip, pb, pa, ir, cb, ca, pb/ip.n(), (pb/ip)/(cb/ic).n(), pb/ip.n() - cb/ic.n() target = target + step n = n + 2