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Block[{a, m = 20, s}, a = ConstantArray[{}, m]; s = {Prime@ PrimePi@ m}; Do[If[# <= m, If[FreeQ[a[[#]], First@ s], a = ReplacePart[a, # -> Append[a[[#]], FirstLast@ s]], Nothing]; AppendTo[s, Last@ s], If[Last@ s == 2, s = DeleteCases[s, 2]; If[Length@ s == 0, Break[], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]], s = MapAt[Prime[PrimePi[#] - 1] &, s, -1]]] &@ Total[s], {i, Infinity}]; Union /@ a // Flatten]

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Michael De Vlieger: Error in program fixed.

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Table plotting prime p in row n at pi(p) place, interposingintervening primes missing from row n are shown by "." as a place holder:

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David A. Corneth: Has the sequence "a(n0 is the largest k such that prime(n) isn't the smallest part of any partition of k in primes." been investigated?

David James Sycamore: Do you mean: 3,7,14,23,38,48,66...? If so see comments in A333266, where the above is a(n) - 1.

David A. Corneth: yes! thanks