Determine all strictly positive integers that are coprime with all numbers of the form $2^{n}+3^{n}+6^{n}-1$, for $n$ a natural number.
We are looking for prime numbers that do not divide any of the $2^{n}+3^{n}+6^{n}-1$ for $n$ a natural integer.

2 is not one of them, because it divides $2+3+6-1$.

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Let $p \geqslant 5$ be a prime. 2, 3, and 6 are coprime with $p$, so $2^{p-1} \equiv 3^{p-1} \equiv 6^{p-1} \equiv 1(\bmod p)$.

$6\left(2^{p-2}+3^{p-2}+6^{p-2}-1\right)=3 \times 2^{p-1}+2 \times 3^{p-1}+6^{p-1}-6 \equiv 3+2+1-6=0(\bmod p)$. Therefore, $p$ divides $6\left(2^{p-2}+3^{p-2}+6^{p-2}-1\right)$, and since $p$ is coprime with 6, $p$ divides $2^{p-2}+3^{p-2}+6^{p-2}-1$.

There does not exist a prime number that does not divide any of the $2^{n}+3^{n}+6^{n}-1$ for $n$ a natural integer, so the only integer prime with all numbers of this form is 1.