cp's OEIS Frontend

Numbers of this form can be represented in the numeral system described in A235860 with k - 1 ones followed to the right by k - 1 twos or k ones followed to the right by k - 1 twos, like this: 1, 12, 112, 1122, 11122, 111222, 1111222, ... (1, 3, 4, 8, 9, 17, 18, ... in decimal) and are the least numbers that need one more digit to be represented than any of their predecessors.

The corresponding sequence of primes starts 3, 17, 67, 1031, 1180591620717411303491, ...

These prime numbers can be written in the numeral system described in A235860 (perhaps not minimally) this way : 2..21..12..2 (or 1..12..2) k twos followed to the right by l ones, followed to the right by m twos.

k can be zero, with the arbitrary limitation, when k = 0, l <= m.

When k = m = 1 we get primes of the form 2^(l + 2) + l + 2.

It must be noted these primes include the Mersenne primes 3, 7, 31, 127, 8191, ... as well as the Fermat primes 3, 5, 17, 257, 65537, with the exception of 5. Mersenne primes can be represented by a one followed to the right by an even number of twos (if the number of twos is odd, the Mersenne number is divisible by 3) with the exception of 3 represented as 12. The Fermat numbers can be represented with three ones followed to the right by a Mersenne number of twos (2^t - 1 (t = 0, 1, 2, 3, 4, 5,...)) : 3 = 111 instead of shorter 12, 5 = 1112 instead of shorter 21, 17 = 111222, 257 = 1112222222, 65537 = 111222222222222222. The composite (divisible by 641) 2^32 + 1 : three ones followed to the right by thirty one twos. The second Fermat prime: 5, appears in this sequence if we let l > m and l <= 3 when k = 0.

By A235860 3, 7 , 17 and 31 can be represented as 12, 122, 111222, 12222 cases when k=0 (primes of the form 2^(m+1) + l - 2: 31 = 2^5 +1 -2). And 11, 73, 191 as 212, 211122, 2122222 (73 = 2^7 - 2^6 + 2^3 + 3 - 2).