The following phenomenon was discovered by Enzo Creti :
This output
? select(m->ispseudoprime(4*10^m+1)==1,[0..500])
%21 = [0, 1, 2, 3, 13, 229, 242, 309]
? select(m->ispseudoprime(16*100^m+1)==1,[0..500])
%22 = [0, 1, 2, 9, 18, 29, 34, 39, 42, 47, 75, 89, 95, 343, 406, 420]
?
shows that in the range $0\le m\le 500$ , there are $8$ primes of the form $$4\cdot 10^m+1$$ while there are $16$ primes of the form $$16\cdot 100^m+1$$ although those numbers should have approximately half of the probability to be prime.
One reason could be that $\ 16\cdot 100^m+1\ $ for $\ m>0\ $ can only have prime factors of the form $\ 4k+1\ $.
If $m$ is even, $\ 4\cdot 10^m+1\ $ with $\ m>0\ $ must have prime factors of the form $\ 4k+1\ $ , but $\ 16\cdot 100^m+1\ $ must have prime factors of the form $\ 8k+1\ $
Can this overcompensate the growth rate, or is this another case of "the law of small numbers" ?