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Ｎθ≔¹ηＷ¬№υθ«≦⊕η≔∨υ⟦⁰⟧ζ≔⟦⟧υＦ⊕θＦζ⊞υ⁺λＸκη»Ｉη

Try it online! Link is to verbose version of code. Outputs the nth value. Brute force, so times out on TIO for n=23 and similar higher values. Explanation:

Ｎθ

Input n.

≔¹η

Start searching for powers.

Ｗ¬№υθ«

Repeat until a one of the sums includes the input.

≦⊕η

Try the next power.

≔∨υ⟦⁰⟧ζ

Save the previous set of sums, or use a sum of zero for the first pass (this is needed to make the code output 2 for an input of zero instead of the 1 that an empty sum would imply.)

≔⟦⟧υ

Start a new set of sums.

Ｆ⊕θ

Loop over all of the integers up to and including n.

Ｆζ

Loop over all of the previous sums.

⊞υ⁺λＸκη

Add the current power to the previous sum and save the result.

»Ｉη

Output the highest power needed.