This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
limit = 226
from functools import lru_cache
sq = [k**2 for k in range(1, int(limit**.5)+2) if k**2 + 3 <= limit]
sqs = set(sq)
@lru_cache(maxsize=None)
def findsums(n, m):
if m == 1: return {(n, )} if n in sqs else set()
return set(tuple(sorted(t+(s,))) for s in sqs for t in findsums(n-s, m-1))
print([n for n in range(4, limit+1) if len(findsums(n, 4)) >= 6]) # Michael S. Branicky, Apr 20 2021
5616 = 1^3 + 8^3 + 12^3 + 15^3
= 2^3 + 8^3 + 10^3 + 16^3
= 4^3 + 4^3 + 14^3 + 14^3
= 4^3 + 5^3 + 11^3 + 16^3
= 8^3 + 9^3 + 10^3 + 15^3
so 5616 is a term.
from itertools import combinations_with_replacement as cwr
from collections import defaultdict
keep = defaultdict(lambda: 0)
power_terms = [x ** 3 for x in range(1, 50)]
for pos in cwr(power_terms, 4):
tot = sum(pos)
keep[tot] += 1
rets = sorted([k for k, v in keep.items() if v >= 5])
for x in range(len(rets)):
print(rets[x], end=", ")