A137389

[Maximilian Hasler]

On Mon, Apr 28, 2008 at 2:22 PM, Alexander Povolotsky
<apovolot at gmail.com> wrote:
>  So it would make sense to revise this sequence with above suggested reduction
>  (division by 4)
>  a(n) = (2^(prime(n)) + 2^(prime(n + 1))) / 2^(prime(1))

A funny way to write 4.
While it is usual a good idea to reduce a (new) sequence by its gcd,
in the current case, the sequence is
anyway "already there" as 2^prime(n)*(1 + 2^(prime_gap(n)).
If I was searching such a sequence and would not find it, noticing
that the terms are divisible by an increasingly higher power of 2, I
would probably have the idea of dividing each term by the highest
possible such power. Thus, IMHO, the sequence (1 + 2^(prime_gap(n)),
i.e.

a(n) = 1 + 2^( prime(n + 1) - prime(n) )
[3, 5, 5, 17, 5, 17, 5, 17, 65, 5, 65, 17, 5, 17, 65, 65, 5, 65, 17,
5, 65, 17, 65, 257, 17, 5, 17, 5, 17, 16385, ...]

is indeed the most useful among all of these.
M.H.