Fermat Primes and Brazilian Numbers

Today I turned $25156$ days old and one of the entries, AO63799, in the OEIS (Online Encyclopaedia of Integer Sequences) for this number states that: $25156$ belongs to the set of numbers $n$ such that n+3, n+5, n+17, n+257, n+65537 are all primes.

At first the numbers $3, 5, 17, 257 \text{ and } 65537$ appeared quite arbitrary but helpfully the comment is made that these numbers are the known Fermat primes. These are primes of the form: $$2^{2^k} + 1, \text{ for some k } >= 0$$ It is conjectured that there are only five values of $k$ that produce such primes, namely $0, 1, 2, 3 \text{ and } 4$ corresponding to $3, 5, 17, 257 \text{ and } 65537$. It has been confirmed that values of k such that $5 \leq k \leq 32$ produce composite numbers.

Of course, these primes are not be confused with the Mersenne primes that are of the form $2^k-1$ and that are probably infinite in number.

In the comments for OEIS AO63799, it's also stated that no Fermat prime is a Brazilian number which of course immediately prompted me to find what defined a Brazilian number. These numbers are listed in OEIS A125134 and defined as:

Numbers $n$ such that there is a natural number $b$ with $1 < b < n-1$ such that the representation of $n$ in base $b$ has all equal digits.

All even numbers $\geq$ 8 are Brazilian numbers because $2p=2(p-1)+2$ is written $22$ in base $p-1 \text{ if } p-1>2$, that is true if $p \geq 4$. The odd Brazilian numbers are listed in OEIS A257521 and are fairly common, with some being prime numbers:

7, 13, 15, 21, 27, 31, 33, 35, 39, 43, 45, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 121, 123, 125, 127, 129, 133, 135, 141, 143, 145, 147, 153, 155, 157, 159, 161, 165, 171, 175, 177, 183, 185, 187, 189, 195, ...

As an example, take 27 in the above sequence which can be expressed as $33_8$. As for the even numbers, take a number like 28. It can be written as 2(14-1)+2 and thus can be represented as $22_{13}$.