Levels (1,i)

[reismann at free.fr]

Dear seqfans,

I had no answer about my "classification of prime numbers" on behalf of seqfans,
what I understand completely because of the indistinctness of my presentation.
Then I am going to present you a precise point of the classification : the
levels (1,i).

Def : Let p(i) denote the i-th prime. If 2p(n) - p(n+1) is a prime, say p(n-i),
then we say that p(n) has level(1,i).
or
Let p(i) denote the i-th prime . If p(n) - g(n) is a prime, say p(n-i),
then we say that p(n) has level(1,i), with g(n) = p(n+1) - p(n).

This primes had a level 1 (A117563) and a weight = p(n) - g(n) (A117078).
For more details and datas you can see :
http://reismann.free.fr/niveaux1i.php

primes of level(1,1) : A006562 (Balanced primes (of order one): primes which are
the average of the previous prime and the following prime.)
Neil rejected my sequence some months ago because A006562 already existed.
I understand even if I would like that my definition appears on the OEIS.
primes of level(1,2) : A117876
primes of level(1,3) : A118467
primes of level(1,5) : A118464
primes of level(1,9) : A119404
primes of level(1,10) : A119403
primes of level(1,11) : A119402
I did not submit all the sequences, that does not seem to me useful.

Ex : p(971579) has level(1,12) :
p(971580) - p(971579) = p(971579) - p(971579 - 12),
p(971580) - p(971579) = p(971579) - p(971567),
15014677 - 15014557 = 15014557 - 15014437 = 120 = 6*20.

The research of the levels(1,i) is very fun and on the first 3000000 terms, I
found (with the help of my friend, Fabien Sibenaler) only one level(1,14) :
p(1945710) - p(1945709) = p(1945709) - p(1945709-14),
p(1945710) - p(1945709) = p(1945709) - p(1945695),
31515557 - 31515413 = 31515413 - 31515269 = 144 = 6*24.

There are two primes of level 1 without sublevel i (exceptions) because the
weight is not a prime number :
p(6)=13, weight = 9, level = 1, g=4.
p(11)=31, weight = 25, level = 1, g=6.

We can make easily some conjectures :
There is an infinite number of primes in every levels(1,i).
The sublevels i are not limited (i -> infinity).

I hope to have comments on this subject.
Have a nice day.

Rémi