I'm a student in Calculus BC, and I was thinking about the density of the primes. I wanted to compare the prime series to other series that are the same density throughout, i.e. multiples of 2, 3, or 4. I created a spreadsheet with the first 100,000 primes, and compared the prime numbers $p_n$ with multiples of $kn$, where k is a natural number, for instance 2. I was interested in the ratio between the 2 sequences, so I looked at $p_n/kn$.
When I graphed $kn$ vs. $p_n/kn$, the result was a logarithmic curve. I looked at the logarithmic regression equation for many of the graphs, and they all had a similar coefficient in front of the $ln(x)$: $3/ke$, where $k$ is the multiple being used in the sequence. The pattern holds up to $k=24$, so I'm fairly certain this is a legitimate discovery.
I haven't been able to figure out the y-intercept for the equations yet, but they have a pattern too. Is there any explanation for why this pattern of $3/ke$ exists?