- The least prime $p$ such that $p+2n$ is also prime: [A020483][7]$(n)$, and the smallest number $x$ such that $\sigma(x+2n) = \sigma(x)+2n$: [A054906][8]$(n)$.    

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

- The number of subwords of length $n$ in the infinite word generated by $a \to aab, \  b \to  b$ : [A006697][3]$(n)$, and the maximal number of distinct nonempty substrings of any binary string of length $n$, plus one: [A094913][4]$(n)+1$.     

- The number of distinct values taken by ${\omega^{\omega^{.^{.^{.^{\omega}}}}}}$ (with $n$ $\omega$'s and parentheses inserted in all possible ways) where $\omega$ is the first transfinite ordinal omega: [A199812][5]$(n)$, and the number of unlabeled rooted trees with at most $n$ nodes, minus $n$ plus one: [A087803][6]$(n)- n + 1$.  






  [1]: https://oeis.org/A084673
  [2]: https://oeis.org/A037055
  [3]: https://oeis.org/A006697
  [4]: https://oeis.org/A094913
  [5]: https://oeis.org/A199812
  [6]: https://oeis.org/A087803
  [7]: https://oeis.org/A020483
  [8]: https://oeis.org/A054906