(3^n+1)/2 is prime. A finite sequence?

[all at abouthugo.de]

SeqFans,

(3^n+1)/2 is prime for n=1,2,4,16,32,64. Can it be shown that no other
(larger) sequence terms exist? The only candidates for this have the
form n = 0 mod 32.
The corresponding primes have a special form:
(3^1+1)/2=2
(3^2+1)/2=5=1+2^2
(3^4+1)/2=41=1+(2^3)*5
[ (3^8+1)/2=3281=1+(2^4)*5*41=17*193 ] composite
(3^16+1)/2=21523361=1+(2^5)*5*17*41*193
(3^32+1)/2=926510094425921=1+(2^6)*5*17*41*193*21523361
(3^64+1)/2=1716841910146256242328924544641=
           1+(2^7)*5*17*41*193*21523361*926510094425921

No more primes of the form (3^(32*m)+1)/2 found up to m=800. Is there
some similarity to the Fermat primes?

Hugo