Stern's diatomic series - frequencies

[Gerald McGarvey]

A very good idea!

The initial frequencies of k in A002487 is related to how soon it shows up 
in A007306.
There are phi(k) odd n's with A002487(n) = k. The sooner k's shows up in 
A007306,
the smaller these odd n's (and therefore 2*n, 2^2*n, etc.) will be. So a 
prime can have a
higher frequency than a larger prime near it (for a while) if its odd n's 
show up significantly
sooner, but eventually the larger prime will have a higher frequency (due 
to a larger phi(k)).

In the case of 71, there are 8 odd n's less than 1024 with A002487(n) = k, 
whereas
for 73 there are only 2, the next prime is 79 which has only 4 (also 67 
only has 4).

What is it about 71 that makes the 71's show up sooner in A007306 than the 
67's, 73's, 79's?

Gerald

At 06:15 PM 9/14/2006, Christian G. Bower wrote:
>No answers here, but an idea.
>
>Look at A007306 (denominators of Farey fractions) which gives the odd
>indices of A002487 and since A002487(n*2^k) = A002487(n) it governs the
>frequency of values in the sequence.
>
>Every k in A007306 appears phi(k) times, (except 1, a special case),
>hence favoring the primes. The frequency in A002487 is probably related
>to how closely clustered those values appear in A007306.
>
>It appears that every value in A002487 has density 0, so I expect the
>most frequent value will continue to increase.
>
>Christian