Crazy Librarian's Arithmetic Sequence of Primes

Pyth, 49 bytes

Like the Python3 and other Pyth answer, the runtime for finding 100 numbers is prohibitive. (test suite gives 10)

?}z"1379".f&!tPZ!|FmtPvjzc`Z]dhl`Z*TT3"Empty Set"

Try it online

Python 3, 188 bytes

x=input()
k=1
i=100
if x in"024568":i=print("Empty Set")
while i:k+=1;s=str(k);i-=all(sum(p%d<1for d in range(2,p))<4for p in[k*int(s[:j]+x+s[j:])for j in range(len(s)+1)])and not print(k)

Badly golfed, but here's something for now. Instead of checking p in P and F(z,p) subset of P, we check that p*f is a semiprime for each f in F(z,p). Combine that with trial division for primality testing, and you get an O(scary) algorithm.

Perl, 124 bytes

$p=prime_iterator;y/1379//or$i=+~print'Empty Set'}while($i<100){$_=&$p;is_prime"$`@F$'"or redo while//g;$i++

Requires the following command line option: -palMntheory=:all, counted as 16. Input taken from stdin.

Uses @DanaJ's Math::Prime::Util module for perl (loaded with the pragma ntheory). Get it with:

cpan install Math::Prime::Util
cpan install Math::Prime::Util::GMP

is_prime is deterministic for all values less than 2⁶⁴, which is sufficient for our purposes.

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Sample Usage

$ echo 2|perl -palMntheory=:all oscary.pl
Empty Set

$ echo 7|perl -palMntheory=:all oscary.pl
3
19
97
433
487
541
691
757
853
1471
.
.
.
718705783
720574573
737773357
745157779
747215167
767717017
768743377
770294977
771778477
774577777

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Expected Runtimes

x = 1 : 1m 09.2s
x = 3 : 0m 04.2s
x = 7 : 2m 52.5s
x = 9 : 0m 11.5s