[math-fun] Primes of the form 1+2*p^k

Dumb questions re primes. Primes of the form 1+2^k are quite rare. ;-) Primes of the form 1+2*3^k seem to be less rare. Primes of the form 1+2*5^k seem to get rarer. Primes of the form 1+2*7^k seem to be quite rare. (I don't have a fast machine, but I'm having trouble finding even one.) Primes of the form 1+2*11^k seem to be less rare. Also, how rare are primes of the form 1+2*p_1*p_2*p_3..., where p_i are odd primes (i.e., primes to the 1st power only) ? (Perhaps these primes should be called "Euclid primes" after Euclid's proof of the infinite # of primes -- if they have no other name?) Anything known about these distributions? Also, is the discrete log particularly cheap to compute for any of these prime forms?