[seqfan] Questions relating to "Sums of three Fermat numbers" -- sums of 5? Primes in A155877? Artificial or not?

[Jonathan Post]

First, I'm grateful to R. J. Mathar for extending my seq A155877  Sums
of three Fermat numbers.

Second, it is a famously interesting question that we don't know if an
infinite number of Fermat numbers A000215 are Fermat primes A019434,
or even which the next one is. Dmitry Kamenetsky commented "It is
conjectured that there are only 5 terms. Currently it has been shown
that 2^(2^n) + 1 is composite for 5<=n<=32 (see Eric Weisstein's
Fermat Primes link,...  Sep 28 2008"

Third, it is obvious that any sum of two Fermat numbers is composite.

Fourth, that's why I submitted A155877, as the next step (sum of 1
unknown, sum of two trivial, sum of 3 scrutinized casually) yet didn't
really know if anyone cared until the extension of Feb 06 2009.

Fifth, I'd commented in the seq that "Primes in this sequence begin
11, 13, 23, 37, 263, 277" because that's what led me to submit.  The
question is, to sefans, is it worth breaking that out and submitting
the sequence "Primes which are sums of three Fermat numbers"?

11, 13, 23, 37, 263, 277, 65543, 65557, ...
and no more through 4294967307.

I honestly can't tell if this is self-indulgent noodling about, or an
interesting and unlikely to yield proofs query combining addition,
multiplication, and exponentiation.  Is is artificial?  Or is it the
next in a logical supersequence of sequences, the next entries being
"Sums of five Fermat numbers" and primes in that. 3 + 5 + 17 + 257 +
65537 is composite, for instance, 65819 = 13 * 61 * 83, and there are
seqfans (I think including njas) who would be annoyed even to call
that number "3-brilliant." But 3 + 17 + 17 + 257 + 65537 = 65831 is
prime, and so forth.

"Sum of a Fermat number of Fermat numbers" seems to me simultaneously
recursive and artificial.  It's an instantiation of a transform
applicable to quite a vast number of seqs, which is almost never
interesting. "Prime sum of a prime number of prime numbers" ad
infinitum.

So I ask, rather than ignore the reasonable plea by njas not just to
make up stuff for its own sake.

Thank you for your opinions, as I'm too close to be objective.

And have a romantic day tomorrow...

Best,

prof. Jonathan Vos Post