In a recent post on math stackexchange, the author came across a book called “Those Fascinating Numbers” by Jean-Marie De Koninck. Intrigued, they decided to take a look at the book and were overwhelmed by the number of integers and facts covered. The author was particularly fascinated by a quote from the preface that stated the probability of the second prime factor of a randomly chosen integer being smaller than 37 is approximately 1/2. They were initially skeptical of this fact but then started to see its plausibility. Curious to find a proof, the author wrote some code in Sage to test it. They discovered that the key idea behind this fact is the use of the Sieve of Eratosthenes to compute the density of numbers whose second prime is a particular prime. Through their calculations, they found that 37 is the median second prime, which further proves the initial statement. The author also mentions the possibility of computing the median third prime and even explores asymptotics for the growth of the median k-th prime as a function of k. Overall, the author found this fact about 37 to be mind-boggling and thoroughly enjoyed exploring it. They express their excitement to share more blog posts
https://grossack.site/2023/11/08/37-median.html