primes per octave?

Hi all-

I have been studying primes a bit in terms of how they appear in the
overtone series. This is the same as the general question of prime
distribution, except the qualification of prime distribution between
powers of two is of especial musical interest. Im sure this has been
explored by number theorists, but i haven't been able to find anything.

For example, the number of primes appearing in each of the first few
octaves is:

2^1: 2
2^2: 2
2^3: 2
2^4: 5
2^5: 7
2^6: 13
2^7: 23
2^8: 43
2^9: 75
2^10: 137
2^11: 254

notice that most of these octaves have a prime number of primes, but
not all of them. whats up with that?

any comments or pointers to info would be appreciated!

Dante

--- In tuning-math@yahoogroups.com, "danterosati" <dante@...> wrote:

> notice that most of these octaves have a prime number of primes, but
> not all of them. whats up with that?

Primes are fairly numerous for small values.

> any comments or pointers to info would be appreciated!

Hoheisel's theorem implies that for any octave [N, 2N] for N above
some value, there will always be a prime number in the octave.
Craner's conjecture says that that lim sup (p_{n+1} - p_n_/ln(p_n)^2 =
1; so that the largest gaps are about ln(p_n)^2 in size. The prime
number theorem allows an estimate of the the number of primes in the
octave [N. 2N] as
li(2N)- li(N) = integral from N to 2N of dx/ln(x).

--- In tuning-math@yahoogroups.com, "danterosati" <dante@...> wrote:

> For example, the number of primes appearing in each of the first few
> octaves is:
>
> 2^1: 2
> 2^2: 2
> 2^3: 2
> 2^4: 5
> 2^5: 7
> 2^6: 13
> 2^7: 23
> 2^8: 43
> 2^9: 75
> 2^10: 137
> 2^11: 254

The handbook of integer sequences doesn't have this, I'd like to add it.