Prime Averages

[Joshua Searle]

SeqFans,

This sequence has no particular purpose but since I spent a little time on it, I thought I'd ask if people would want it added to the OEIS.

The sequence is generated via the following:

-Start with an empty list.

-Take the average of the nth prime number p_n with itself for all p_n - this is trivially the prime numbers - and add them to the list.

-The smallest number generated this way is 2 (from 2,2).

-Now take the average of p_n with p_n+1 in a similar fashion. The smallest number generated this way is 4 (from 3,5). Ignore when an average isn't an integer, which only happens when 2 is one of the primes.

-Continue with p_n+m as m->inf

Alternatively, you can start with all the integers and treat it as a sieve.

The smallest number generated after each "pass" is:

2,4,8,25,28,22,49,46,203,136,184,146...

The number of possible averages strongly suggests that there are no 'holes' though this may well be difficult to prove (similar to Goldbach?). The smallest number that has not appeared after each pass is:

4,8,22,22,22,46,46,136,136,146,146...

For the first sequence, it is not monotonically increasing. If a decreasing step is 0 and an increasing step 1, what does the running average converge to? I can see it converging to 0.5, but some other number >0.5 is also plausible at least at first glance.

From the initial terms, it seems to grow at roughly O(n^2) - to which I note that the smallest number Eratosthenes sieve eliminates on each pass is exactly n^2.

Joshua