All the way random, as it turns out, as 17 has been officially marked as the most random number between 1-20, by no less an authority than William Feller, a mathematician who taught at Princeton. Later, he died. Arnold Zwicky, who studied mathematics at Princeton, wrote about 17 several times, and even presented a proof of its randomness:
"A completely random number can't (should not) be too large. Say, less than or equal to 20. After that it's a breeze. Working from below, it clearly can't be 1, nor can it be even, and hence neither 2, 4, 6, 8, 10, 12, 14, 16, 18, nor 20. 3 is for the Trinity and 5 is too important in base 10 notation; 7 and 11 are lucky, and 9 is a perfect square and hence hardly random. 13 is unlucky; 15 is a multiple of 5; and 19 is too close to 20. That leaves 17 as the only possible candidate. Q.e.d."
So there's that, and although there are plenty of cultural signals for 17 (many by Abba), the number by itself doesn't get a lot of play in numerological circles or society at large. However, 17 does have its share of claims to fame. Notably, it is one of several numbers closely associated with Eris, the Greek goddess of discord and strife. Eris is most famous for tossing the apple that started that Trojan War, and slightly less well-known for fighting Wonder Woman.
And while that's kind of a crap association, things could be a lot worse for 17. It could be 23, another number noted for randomness, the number most closely associated with Eris, and a number that may never live down the ignominy of being the subject of an awful movie about Jim Carrey peeing on things and calling it synchronicity. Joel Schumacher's The Number 23 2: The Number 17 probably won't be coming to a theater near you, but that's just the kind of random and unfortunate nonsense 17 is infamous for.
For my part, I tend not to notice the number 17 unless it's in conjunction with an incongruous element, but that might be appropriate. My favorite of these is from the April, 2002 issue of "Seventeen," with a photo of the lovely Keri "Felicity" Russell, oblivious to the fact that the cover copy was going to ask the eternal question, "Is that my butt?"
It made me laugh in April of 2002, and it makes me laugh today.