Terms are conjectural, even under the Riemann Hypothesis.
(1) The initial term a(1)=0 gives a simple restatement of Legendre's conjecture: There are no primes between 0^2 and 1^2, but there is a prime between m^2 and (m+1)^2 for m>0.
(2) Lists of known maximum prime gaps and known first occurrences of prime gaps help verify the initial terms in this sequence. However, a lengthy computation would be needed for subsequent terms.
Table of n, a(n) for n=1..5.
T. R. Nicely, List of prime gaps
Tomás Oliveira e Silva, Gaps between consecutive primes
The second term is a(2)=1051 because there are no primes between 1051^(3/2) and 1052^(3/2), but there is at least one prime between m^(3/2) and (m+1)^(3/2) for m>1051.
Cf. A144140, A192870.
Sequence in context: A020389 A252609 A185680 * A090005 A225714 A158692
Adjacent sequences: A191855 A191856 A191857 * A191859 A191860 A191861
Alexei Kourbatov, Jun 18 2011