no. n = 4, i = 2 is a counter-example.

yes, because for any i < n:

The sequence of mersenne numbers M_p = 2^p -1 (here p shall only take on prime numbers) is a primitive sequence?

yes, becasue for i < n we have:The sequence of Fermat numbers 2^(2^n) +1 is a primitive sequence?

i'd be very surprised if the answer to this one is "yes"! have you checked it for some values of n probably using maple or something?(Hard question) The sequence of numbers defined by a_n = 2*3*5....*p_n -1 (p_n is the nth prime number) is a primitive sequence.

does anybody here know if a sequence with this property that for all has any name or not?