a(n)>3.6*10^24 for n >=4. This lower bound was established by exhaustive crowd computing.
Upper bounds:
a(4)<= 3915335521240189321820073984
=467175960^3+1562319144^3
=569783808^3+1550898288^3
=1085420968^3+1381483928^3
=1157553216^3+1332193392^3
a(5)<=
1508202165690304620654479410485250200981504
=12156471201588^3+114634141265868^3
=33798481396036^3+113692639229372^3
=64341822928208^3+107486686558048^3
=69397851840576^3+105492106344912^3
=72561474196232^3+104039736026296^3
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