The n chosen integers need not be distinct.
By "more than half of all integers" we mean more precisely "more than half of the integers in m..m, for all sufficiently large m (depending on n)", and similarly with 1..m for "more than half of all positive integers".
Equivalently, a(n) is the least prime p such that more than half of all positive integers can be written as a product of primes of which n or more are not greater than p. (In this sense, a(n) might be called the median nth least prime factor of the integers.)
The number of integers that satisfy the "product of primes" criterion for p = prime(m) is the same in every interval of primorial(m)^n integers and is A281891(m,n). Primorial(m) = A002110(m), product of the first m primes.
a(n) is the least k = prime(m) such that 2 * A281891(m,n) > A002110(m)^n.
a(n) is the least k such that more than half of all positive integers equate to the volume of an orthotope with integral sides at least n of which are orthogonal with length between 2 and k inclusive.
The next term is estimated to be a(5) ~ 3*10^18.
