[quote=R.D. Silverman;140306]Although not explicitly stated, I believe that the domain is N. Now,
f(x) is prime for x = 0, 4 and no other. If you accept the more general definition of prime (i.e. not restricted to just N) then f(x) will be prime i.o. (although a proof is lacking). If we allow x \in R, then f(x) is indeed prime the required number of times.[/quote] If we allow real numbers as x, then f(x) is prime 19 times  9 times for negative x values as f(x) increases, once for x=0 where f(x) levels off at 29, and 9 more times for positive x values as f(x) decreases. There would be 10 unique primes, but 19 values of x that produce a prime f(x). 
[QUOTE=R.D. Silverman;140306]Although not explicitly stated, I believe that the domain is N. Now,
f(x) is prime for x = 0, 4 and no other. If you accept the more general definition of prime (i.e. not restricted to just N) then f(x) will be prime i.o. (although a proof is lacking). If we allow x \in R, then f(x) is indeed prime the required number of times.[/QUOTE]Thanks, yeah I was assuming x is an element of R. f(x)=398x^2; x element of N To satisfy MiniGeek's requirement of total primes: f(x)=143x^2; x element of Z 
I am obviously not communicating well. However you could have referred me to primes.utm.edu (as done by Cruelty) or Wikipedia or some other helpful web site. I am looking for "interesting" examples as listed on these web sites.
On the primes.utm.edu site Rudolf Ondrejka lists ten rare primes. One example he refers to as a beastly palindrome of the type (10^n 666)*10n2+1. Has it been proven that only 7 exist? Wikipedia, states that there is only one positive Genocchi prime; has this been proven? 
[QUOTE=Housemouse;140313]However you could have referred me to primes.utm.edu (as done by Cruelty) or Wikipedia or some other helpful web site.[/QUOTE]It is usually assumed that you know how to [url=http://justfuckinggoogleit.com/]google[/url] things you are interested in.

[QUOTE=Housemouse;140313]On the primes.utm.edu site Rudolf Ondrejka lists ten rare primes. One example he refers to as a beastly palindrome of the type (10^n 666)*10n2+1.
Has it been proven that only 7 exist?[/QUOTE] No. Given that (10^n + 666) * 10^(n2) + 1 is not divisible by 2, 3, or 5, a quick guess at the 'chance' it's prime as 15/4 * (1/log(10^(2n2))) The sum of this from 2 to 3000 is 6.98, so having 7 from n = 2 to 3000 is pretty much what you'd expect. The expected number up to a million is 11.72, so it would be unusual if only 7 existed. In fact, since the harmonic series diverges, you'd naively expect an infinite number of such primes. [QUOTE=Housemouse;140313]Wikipedia, states that there is only one positive Genocchi prime; has this been proven?[/QUOTE] MathWorld has "D. Terr (pers. comm., Jun. 8, 2004) proved that these are in fact, the only prime Genocchi numbers.". 
To the original question: Of course there are uncountably many sets of primes (beth_1), a countable number of which sets are finite; but the question seems to be about intuitively 'interesting' sets of primes.
Toward that end I suggest my small compilation here: [url]http://en.wikipedia.org/wiki/User:CRGreathouse/Tables_of_special_primes[/url] 
Additional "interesting examples"
I found additional "interesting examples" at primes.utm.edu, under Rudolf Ondrejka's top ten.
Sometimes finding useful information using google is like looking for a needle in a hay stack. 
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