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

[Jonathan Post]

"There exist a variety of formulas for either producing the nth primeas a function of n or taking on only prime values. However, all suchformulas require either extremely accurate knowledge of some unknownconstant, or effectively require knowledge of the primes ahead of timein order to use the formula (Dudley 1969; Ribenboim 1996, p. 186)."
Weisstein, Eric W. "Prime Formulas." From MathWorld--A Wolfram WebResource. http://mathworld.wolfram.com/PrimeFormulas.html
One can construct sequences where "nexprime" is replaced by variousfunctions built from approximations to the prime counting function isthe function pi(x), which gives the number of primes less than orequal to a given number x.
First, "one of the most fundamental and important results in numbertheory is the asymptotic form of pi(n) as n becomes large. This isgiven by the prime number theorem, which states thatpi(n)∼Li(n),where Li(x) is the logarithmic integral and ∼ is asymptotic notation.This relation was first postulated by Gauss in 1792 (when he was 15years old), although not revealed until an 1849 letter to Johann Enckeand not published until 1863 (Gauss 1863; Havil 2003, pp. 176-177)."
Then various corrections to Li(n), as summarized in:
Weisstein, Eric W. "Prime Counting Function." From MathWorld--AWolfram Web Resource.http://mathworld.wolfram.com/PrimeCountingFunction.html
The goal being to have a supersequence of sequences which are provablyasymptotcally almost what Andrew Plewe first asked for.
-- Jonathan Vos Post
-- Jonathan Vos Post