Count the distinct prime factors and the divisors of even numbers between twin primes; see https://math.stackexchange.com/questions/3490592.

Question3490592.java

import java.util.Arrays;

public class Question3490592 {
    final static int n = 0x40000000; // highest number to test

    static boolean [] prime = new boolean [n / 2]; // prime [i] : is 2i + 1 prime?
    
    public static void main (String [] args) {
        Arrays.fill (prime,true);
        
        // calculate the primes we need
        int limit = (int) Math.sqrt (n); // highest factor to test
        for (int p = 3;p <= limit;p += 2) // loop over odd integers
            if (prime [p >> 1])            // only test primes p
                for (int k = 3*p;k < n;k += 2*p) // loop over odd multiples of p
                    prime [k >> 1] = false;      // sieve them out
        
        double totalDivisors = 0;
        long totalPrimeFactors = 0;
        double pairDivisors = 0;
        long pairPrimeFactors = 0;
        long numberCount = 0;
        long pairCount = 0;
        
        for (int p = 3;p < 10000000;p += 2) {
            int even = p + 1;
            int sqrt = (int) Math.sqrt(even);
            int divisorCount = 0;
            int primeFactorCount = 0;
            for (int d = 1;d <= sqrt;d++)
                if (even % d == 0) {
                    divisorCount++;
                    if (d == 2 || ((d & 1) == 1 && prime [d >> 1]))
                        primeFactorCount++;
                    int s = even / d;
                    if (s == 2 || ((s & 1) == 1 && prime [s >> 1]))
                        primeFactorCount++;
                }
            divisorCount <<= 1;
            if (sqrt * sqrt == n)
                divisorCount--;
            
            double divisorLog = Math.log(divisorCount);
            
            totalPrimeFactors += primeFactorCount;
            totalDivisors += divisorLog;
            numberCount++;

            if (prime [p >> 1] && prime [(p >> 1) + 1]) {
                pairPrimeFactors += primeFactorCount;
                pairDivisors += divisorLog;
                pairCount++;
            }
        }
        
        System.out.println("average prime factor count: " + totalPrimeFactors / (double) numberCount);
        System.out.println("between prime factor count: " + pairPrimeFactors / (double) pairCount);
        System.out.println("average disivor count logarithm: " + totalDivisors / (double) numberCount);
        System.out.println("between disivor count logarithm: " + pairDivisors / (double) pairCount);
    }
}