Most "compact" sequence such that there is at least one prime between a(n) and a(n+1)

[Maximilian Hasler]

> What would be the minimum required exponent (not theoretically, but from numerical evidence?).

googling "prime number gaps bertrand conjecture" or so I found on
http://www.physicsforums.com/archive/index.php/t-125719.html:
"For example, if n is large enough, we can guarantee a prime in [n,n+n^0.525]."

but without reference. In http://math.univ-lille1.fr/~ramare/Maths/gap.pdf

it is proved that ] x-x/D , x [ contains a prime number
for D = 28 314 000 and all x >= 10 726 905 041.

This is of course much better.
I just want to repeat that all of these results will *never* enable us
to prove that a given proposal for the sequence (as specified in the
"clarification") is "minimal" or "most compact" in the sense given
earlier, and I'd even dare to say that it will be rather quite easy to
*disprove* minimality for any such "mechanically constructed" sequence
(e.g. by adding "-1" to all terms following some a(n), or so).

Maximilian