This is my first post here. I am a musician, and not a mathematician, but I enjoy doing things to prime numbers and seeing what comes out.
I have defined a sequence which takes the following values for $n$:
- -1 if $n$ is prime
- 1 if $n$ is a practical number
- 0 if $n$ is neither or both
I have then taken a sequence of its partial sums. The first 50 terms are 1,1,0,1,0,1,0,1,1,1,0,1,0,0,0,1,0,1,0,1,1,1,0,1,1,1,1,2,1,2,1,2,2,2,2,3,2,2,2,3,2,3,2,2,2,2,1,2,2,2.
The plot for $n<100000$ looks quite linear:
In order to see quite how linear it was, I then divided each term of the sequence by $n$ and got this plot:
It seems to me like it wants to converge to some value. The arithmetic mean of the last 100 terms is 46.3225.
I vaguely understand that there are some analogies between practical numbers and primes. I am wondering how difficult it would be to establish if the above sequence does in fact converge, and if so, then to what value. I have tried it with other prime-like sequences, such as ludic numbers and lucky numbers, but the other ones didn't seem as neat...