Problem: Determine all positive integers relatively
prime to all the terms of the infinite sequence
\[ a_n = 2^n+3^n+6^n-1, \quad n \ge 1. \]

Solution: The answer is $1$ only (which works).

It suffices to show there are no primes.
For the primes $p=2$ and $p=3$, take $a_2=48$.
For any prime $p \ge 5$ notice that
\begin{align*}
  a_{p-2} &= 2^{p-2} + 3^{p-2} + 6^{p-2} - 1 \\
  &\equiv \frac 12 + \frac 13 + \frac 16 - 1 \pmod p \\
  &\equiv 0 \pmod p
\end{align*}
so no other larger prime works.