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.
a(3) = 1521 = 1^3 + 2^3 + 8^3 + 10^3 = 1^3 + 4^3 + 5^3 + 11^3 = 4^3 + 6^3 + 8^3 + 9^3.
a(2) = 6834 = 1^4 + 2^4 + 4^4 + 9^4 = 3^4 + 4^4 + 7^4 + 8^4. a(3) = 16578 = 1^4 + 2^4 + 9^4 + 10^4 = 2^4 + 5^4 + 6^4 + 11^4 = 3^4 + 7^4 + 8^4 + 10^4.
M:= 4000: # for terms <= M
V:= Vector(M):
for a from 1 while 4*a^3 < M do
for b from a+1 while a^3 + 3*b^3 < M do
for c from b+1 while a^3 + b^3 + 2*c^3 < M do
for d from c+1 do
v:= a^3 + b^3 + c^3 + d^3;
if v > M then break fi;
V[v]:= V[v]+1
od od od od:
select(t -> V[t] > 1, [$1..M]); # Robert Israel, Oct 23 2023
For n = 2: 1729 = 1^3 + 12^3 = 9^3 + 10^3. For n = 3: 5104 = 1^3 + 12^3 + 15^3 = 2^3 + 10^3 + 16^3 = 9^3 + 10^3 + 15^3.
a(2) = 4062500 = 1^5 + 14^5 + 16^5 + 19^5 = 5^5 + 10^5 + 15^5 + 20^5. a(3) = 1479604544 = 3^5 + 48^5 + 52^5 + 61^5 = 13^5 + 36^5 + 51^5 + 64^5 = 18^5 + 36^5 + 44^5 + 66^5.
a(2) = 160426515 = 1^6 + 3^6 + 19^6 + 22^6 = 1^6 + 10^6 + 15^6 + 23^6. a(3) = 1885800643779 = 1^6 + 34^6 + 49^6 + 111^6 = 7^6 + 43^6 + 69^6 + 110^6 = 18^6 + 25^6 + 77^6 + 109^6.