A quite computationally challenging integer sequence I know is the sequence of odd positive integers $n$ for which $\sigma(n)-2n=6$. Here $\sigma(n)$ denotes the sum of positive divisors of $n$. Only 8 terms of this sequence are known, and the largest one, 815634435, was computed on 2011 (sequence [A141548][1]). Larger terms of this sequence could be related to [an eventual proof][2] of the existence of odd weird numbers, an old open question raised by Erdös and Benkoski. 


  [1]: https://oeis.org/A141548
  [2]: https://arxiv.org/pdf/1803.00324.pdf