I have found that a finite sequence, $S_n$ exists describing the differences between integers starting from $P_n$ (the $n$th prime) such that their lowest prime factor is $P_n$. These sequences, starting from ${P_n}^2-P_n$, have length equal to the $n$th term from https://oeis.org/A005867. Each term can be written as $2k\cdot P_n$, where $k$ is some positive integer. By plotting the frequencies of each term against their value, it is clear to see how the mean value of $S_n$ equals $\frac{1}{F_n}$ (from my post at http://www.primes.esy.es/). However, the sets of $k$ values for each sequence $S_n$ are strange. I have written a program to generate $S_n$, however due to the rapidly increasing number of terms it can only go up to $n=7$ ($P_n = 17$) before it takes longer than a few minutes. The $k$ values from $n=1$ to $7$ are as follows:
$1:1$
$2:1$
$3: 1, 2$
$4: 1, 2, 3$
$5: 1, 2, 3, 4, 5$
$6: 1, 2, 3, 4, 5, 6, 7$
$7: 1, 2, 3, 4, 5, 6, 7, 8, 9, 11$
Notice how the case $n=7$ is missing a 10? My question is why is it missing a 10, is there a way to predict the next sequence of $k$ values and even perhaps the order of $S_n$? Combining a large enough number sequences would allow the prediction of all prime numbers up to any value as they would contain every non-prime. Another thing I haven't been able to connect is why the number of terms in the sequence $S_n$ is equal to $\varphi\left(\prod_{i=1}^{n}P_n\right)$, so if someone could suggest a reason that would be really appreciated.
EDIT: I have found that $\sum_{a=1}^{n}S_a=P_{n-1}\#={F_n}^{-1}\cdot\varphi\left(\prod_{i=1}^{n}P_n\right)$, where $P_n\#$ is $P_n$ primorial. Not sure if this helps but it might!
Thanks :)