This is a family of increasingly restrictive Andrica-like conjectures that all imply Legendre's conjecture.
Table of n, a(n) for n=1..4.
Example: For k = 1, one needs n > 30 for the inequality to obtain, and it is conjectured that it holds for all n > 30. In words, the first such inequality says that we expect to see a new prime p(n+1) between p(n) and p(n)+ Sqrt(p(n)).
Sequence in context: A110612 A290363 A161740 * A210431 A296919 A010946
Adjacent sequences: A161620 A161621 A161622 * A161624 A161625 A161626
Daniel Tisdale, Jun 15 2009