TITLE: Sums of squares of primes
QUESTION [7 upvotes]: Question: What is the least number that is a sum of three squares of primes in exactly six ways?

... I know it is not research mathematics. Happy new year!
EDIT: Now that it is answered I should note that I learned this puzzle from a tweet by Ed Southall. I thought it is fun to share.

REPLY [5 votes]: The sequence $f(n)$ given by those smallest integers which can be written in $n$ ways as the sum of squares of three primes has first eleven values,
$$12,219,363,699,1179,2019,2259,3891,4059,6459,5379.$$
As one can see, the $10$th value, $6459$, is larger than the $11$th value, $5379$. This is OEIS A214512, which indicates that it was really T.D. Noe  who observed this earlier.
In any case it will be some time before this puzzle can be used like this again.