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The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

allocated for Gus WisemanHeinz numbers of integer partitions that are the vertex-degrees of some set system with no singletons.

1, 4, 8, 12, 16, 18, 24, 27, 32, 36, 40

1,2

A set system is a finite set of finite nonempty sets.

Each term paired with its Heinz partition and a realizing set system:

1: (): {}

4: (11): {{1,2}}

8: (111): {{1,2,3}}

12: (211): {{1,2},{1,3}}

16: (1111): {{1,2,3,4}}

18: (221): {{1,2},{1,2,3}}

24: (2111): {{1,2},{1,3,4}}

27: (222): {{1,2},{1,3},{2,3}}

32: (11111): {{1,2,3,4,5}}

36: (2211): {{1,2},{1,2,3,4}}

40: (3111): {{1,2},{1,3},{1,4}}

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

hyp[m_]:=Select[mps[m], And[And@@UnsameQ@@@#, UnsameQ@@#, Min@@Length/@#>1]&];

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];

Select[Range[20], !hyp[nrmptn[#]]=={}&]

Cf. A000070, A000569, A056239, A112798, A283877, A306005, A318361, A320922, A320923, A320924, A320925, A321176.

allocated

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Gus Wiseman, Oct 29 2018

approved

editing

allocated for Gus Wiseman

allocated

approved