5, 10, 13, 17, 20, 25 ... are numbers that are the sum of 2 distinct nonzero squares (A004431-let's call it SEQ).

Now, some tasks for you:

1. 25,40,52,73,89 ...; add 2 more numbers in this sequence.
2. In the sequence SEQ find the 1st appearance of 3 consecutive numbers.
3. Find the index of the 1st 4-digit member in SEQ.
4. There is a "run" of 4 consecutive non members of SEQ between 45 and 50 .
a) find a longer "run".
b) find the longest "run" within the 1st 12000 members of SEQ.
5. Find the lowest pandigital member of SEQ.
6. Find a pandigital member(s?) of SEQ, equalling A²+ B² such that the concatenation of the numbers A & B is also pandigital.

Rem: If you have solved 4 or more out of 6 listed tasks please rate this post - I've spent about 2.5 hours to create it.