1. The least prime $p$ such that $p+2n$ is also prime: [A020483][1]$(n)$, and the smallest number $x$ such that $\sigma(x+2n) = \sigma(x)+2n$: [A054906][2]$(n)$.    

2. The smallest prime in which a digit appears $n$ times: [A084673][3]$(n)$, and the smallest prime containing exactly $n$ $1$'s: [A037055][4]$(n)$, for $n>1$.   

3. The number of subwords of length $n$ in the infinite word generated by $a \to aab, \  b \to  b$ : [A006697][5]$(n)$, and the maximal number of distinct nonempty substrings of any binary string of length $n$, plus one: [A094913][6]$(n)+1$.     

4. The number of distinct values taken by ${\omega}$^${\omega}$^${\dots}$^${\omega}$ (with $n$ $\omega$'s and parentheses inserted in all possible ways) where $\omega$ is the first transfinite ordinal omega: [A199812][7]$(n)$, and the number of unlabeled rooted trees with at most $n$ nodes [A087803][8]$(n)$, minus $n$ plus one: [A255170][9]$(n)$.  

5.  The number of transitive permutation groups of degree $n$: [A002106][10]$(n)$ is conjectured to be the number of Galois groups for irreducible polynomials (over $\mathbb{Q}$) of order $n$ (such groups are transitive).  It is a particular case of the [Inverse Galois problem][11].


  [1]: https://oeis.org/A020483
  [2]: https://oeis.org/A054906
  [3]: https://oeis.org/A084673
  [4]: https://oeis.org/A037055
  [5]: https://oeis.org/A006697
  [6]: https://oeis.org/A094913
  [7]: https://oeis.org/A199812
  [8]: https://oeis.org/A087803
  [9]: https://oeis.org/A255170
  [10]: https://oeis.org/A002106
  [11]: https://en.wikipedia.org/wiki/Inverse_Galois_problem