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
13104 is a term because 13104 = 1^3 + 10^3 + 16^3 + 18^3 = 1^3 + 11^3 + 14^3 + 19^3 = 2^3 + 9^3 + 15^3 + 19^3 = 4^3 + 6^3 + 14^3 + 20^3 = 4^3 + 9^3 + 10^3 + 21^3 = 5^3 + 7^3 + 11^3 + 21^3 = 8^3 + 9^3 + 14^3 + 19^3.
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, 1000)]
for pos in cwr(power_terms, 4):
tot = sum(pos)
keep[tot] += 1
rets = sorted([k for k, v in keep.items() if v >= 7])
for x in range(len(rets)):
print(rets[x])
selQ[n_] := Length[ Select[ PowersRepresentations[n, 4, 2], Times @@ # != 0 &]] >= 8; Select[ Range[300], selQ] (* Jean-François Alcover, Oct 03 2013 *)
N:= 1000: # for terms <= N G:= mul(1+x^(i^2)*y, i=1..floor(sqrt(N))): G4:= series(coeff(G,y,4),x,N+1): A:= select(t -> coeff(G4,x,t) >= 7, [$1..N]); # Robert Israel, Nov 19 2023
With[{nn=25},Select[Select[Tally[Total/@Subsets[Range[nn]^2,{4}]],#[[2]]> 6&][[All,1]]//Union,#<=(nn^2-14)&]] (* Harvey P. Dale, Jun 21 2021 *)