Is it a Pascal Prime?
Wolfram Language (Mathematica), 45 bytes

CompositeQ@#&amp;&amp;Binomial~Array~{#-1,#}~FreeQ~#&amp;


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Every composite number n appears exactly twice on row n and cannot appear afterwards. So the condition for Pascal primes is that they don't appear at all within the first n-1 rows.

As far as I can tell, this is shorter than checking that it appears exactly twice within the first n rows and being able to use !PrimeQ instead.