Your guess is off by a bit...

Use the Prime Number Theorem to estimate how many primes are in the set of integers up to 10^100. Then do any of the following.

(1) Estimate how long to store that sequence, assuming you can store one per chip cycle, using say 10^5 chips each operating at 10^12 cycles per second.

(2) Compare the number to estimated atoms in the universe.

(3) Rethink the computation you are doing to see if there is a variant that is computationally feasible. The one stated is not.