This is closely related to this question, in which I asked about pairing of numbers in the same problem, which I defined as follows:
Given a number $n$, the sequence of "prime-adjacent multiples" is defined as any multiple $k$ that is $\pm 1$ away from a prime number.
It essentially boils down to: How frequently do multiples of my number live next to a prime number?
This can be calculated by running an extremely simple python script (as follows, just if you wanted to try this yourself):
import sympy import numpy as np c =  for i in range(1000): c.append(0) for j in range(5000): if sympy.isprime(i*j - 1) or sympy.isprime(i*j + 1): c[-1] += 1 ind = np.argmax(c) print(ind, c[ind])
Which will calculate the number of times a multiple of a number $n$ is adjacent to a prime for each $n$ from 0 to 1000, for the first 5000 multiples.
Running this shows that 30 has the most "prime-adjacent multiples" for the first 5000 multiples. As there was a bit of preamble, I'll restate: I want to know why this is. Why are multiples of 30 adjacent to prime numbers more often than any other number?
Not entirely related:
This may beg the question of how do I know 30 has prime-adjacent multiples "more often than any other number"? It's mainly down to observation, by plotting the count of prime-adjacent multiples for all numbers between 0 and 20,000 (this time for the first 1000 multiples, to save on computation time) gives the following graph:
Where the initial peak at the start is 30 (closely followed by 6). The graph would indicate that no number would ever be able to reach a peak similar in size to the numbers at the start.