For a nonnegative integer m, pi(m) = A000720(m). It is well-known that if
m >= 17, then m/log(m) < pi(m). [Rosser and Schoenfeld]
Fix a real exponent d > 0. If m is big enough, then m < (m/log(m)) + (m/log(m))^(1 + d). In particular, choosing d = 1/n, with n >= 1, we deduce that a(n) exists.
Note that different choices of the exponent d will produce analogous sequences.
The estimates of pi(m) in [Dusart, Thm. 5.1] and [Axler, Thm. 2] allow us to obtain upper and lower bounds for a(n). In particular, we can conclude that in base 10:
a(11) has 20 digits, starting with 430;
a(12) has 22 digits, starting with 826;
a(13) has 25 digits, starting with 1729;
a(14) has 27 digits, starting with 392;
a(15) has 29 digits, starting with 962;
a(16) has 32 digits, starting with 2534.
The tool primecount [Walisch], used to compute pi(10^27) in A006880, can handle pi(m) for m <= 10^31, and since (a(n)) is monotonically increasing, it seems that the computation of a(n) for n >= 16 will be challenging.