sequences analogous to Fortunate numbers

[Frank Buß]

Hi,

I'm a computer programmer and hobby mathematican (without a study) and I
have found some sequences, analogous to Fortunate primes
(http://pauillac.inria.fr/algo/banderier/Computations/prime_factorial.html).
Fortune's conjecture: If you multiply the first n primes, the distance to
the next prime of this product is 1 or a prime, for every n > 0. On some web
pages the next prime, from which you calculate the difference, have to be
greater than the product+1 (sequence A005235).

It looks like there are some other products, for which the distance to the
next prime is a prime. Some simple examples are my following sequences:
A067362-A067365 (referenced by
http://www.primepuzzles.net/conjectures/conj_028.htm), which are
enhancements of A057016 or the sequence A066889 with Fibonacci numbers.
There are some other such sequences, for example with binomial products. I
can't prove, that all sequence numbers are prime, but perhaps someone on
this mailing list can do this?

A much more interesting sequence, I think, is A067836 (referenced by
http://www.primepuzzles.net/conjectures/conj_fb.htm, will be conjecture 29
this saturday): The recursive sequence a(n), starting with n=2: Let
f(n)=f(n-1)*a(n-1) with f(1)=1, a(n)=nxtprime(f(n)+1)-f(n) and nxtprime(x)
defined as returning the smallest prime > x. It is so interesting, because
it looks like every element is prime and there are no duplicates. Now I
don't think that all primes will be produced, because I draw the sorted
prime indexes (temporarly available at
http://www.frank-buss.de/prime/primedelta2.png) until n=262 and it looks
like a smooth curve.

Likewise the first sequences I can't prove this sequence. But I have another
conjecture: It looks like if the distance to the next prime is a prime, then
the distance to the previous prime is also a prime. I have tested it for the
Fortunate primes and for A067836 (see A068192). Perhaps this special
conjecture is easier to prove and helps proving the more general
conjectures. Does someone know other prime delta sequences? If yes, is the
prev prime conjecture true for this sequences?

Regards

Frank

* Programmers are tools for converting caffeine into code.