[sup]My typing is correct. (Anyway, I think it unlikely that the solution is as complex as cyclic numbers.)
[/sup]
In fairness I have actually seen questions in quizes that refer to cyclic numbers as well as things like the next Mersenne prime in a sequence, perfect numbers and triangular numbers. It's not any harder than them in a way. Strangely enough I had to use the sequence in a course on representation theory.
Problem is if you don't put enough numbers in the sequence, it doesn't narrow it down enough to what people might plausibly expect (still many sequences it could be). As for the claimed sequence, with 23 following 15, that's actually listed on sloane, just it doesn't elaborate a great deal.