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:
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: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.
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 \(22_{13} \).
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