all possible pairs

LaTeX4Web 1.4 OUTPUT In this section we study given n how many pairs of primes make n satisfy Condition 1. We have found computationally all the possible pairs of primes that satisfy Condition 1 with a given n £ 3000. The following figure shows the data for n up to 3,000.
LaTeX4Web 1.4 OUTPUT We note four main tendencies. The one with the greatest slope corresponds to the one formed with prime numbers and prime powers. This is explained with the first proposition explained in the section "An open problem". Because only one prime is needed to satisfy Condition 1, the other prime can be any prime smaller than n. Therefore, this first tendency follows the function n/log(n), which describes the distribution of prime numbers. The second greatest slope is formed with even numbers that satisfy Condition 1 with 2. The third one is formed by numbers that satisfy Condition 1 with 3 and the following one with numbers that satisfy Condition 1 with 5. In order to fit a function for each curve, we approximated the function n/log n for each branch with Matlab, and we obtained the following functions:
LaTeX4Web 1.4 OUTPUT The following plot shows a plot of each separated branch with its corresponding curve.

Here we provide the code that we used to compute how many different pairs make n satisfy Condition 1.

AllPossiblePairs.txt AllPossiblePairs.txt
Size : 3.136 Kb
Type : txt

Here you can see an example of the output of the code until n=21.