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A320464 MM-numbers of labeled multi-hypergraphs with no singletons spanning an initial interval of positive integers.

%I A320464 #5 Oct 15 2018 22:21:43
%S A320464 1,13,113,169,377,611,1291,1363,1469,1937,2021,2117,2197,3277,4537,
%T A320464 4859,4901,5249,5311,7423,7943,8249,8507,10933,12769,16211,16403,
%U A320464 16559,16783,16837,17719,19097,20443,20453,24553,25181,25477,26273,26969,27521,28561,28717
%N A320464 MM-numbers of labeled multi-hypergraphs with no singletons spanning an initial interval of positive integers.
%C A320464 A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.
%H A320464 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>
%e A320464 The sequence of terms together with their multiset multisystems begins:
%e A320464      1: {}
%e A320464     13: {{1,2}}
%e A320464    113: {{1,2,3}}
%e A320464    169: {{1,2},{1,2}}
%e A320464    377: {{1,2},{1,3}}
%e A320464    611: {{1,2},{2,3}}
%e A320464   1291: {{1,2,3,4}}
%e A320464   1363: {{1,3},{2,3}}
%e A320464   1469: {{1,2},{1,2,3}}
%e A320464   1937: {{1,2},{3,4}}
%e A320464   2021: {{1,4},{2,3}}
%e A320464   2117: {{1,3},{2,4}}
%e A320464   2197: {{1,2},{1,2},{1,2}}
%e A320464   3277: {{1,3},{1,2,3}}
%e A320464   4537: {{1,2},{1,3,4}}
%e A320464   4859: {{1,4},{1,2,3}}
%e A320464   4901: {{1,2},{1,2},{1,3}}
%e A320464   5249: {{1,3},{1,2,4}}
%e A320464   5311: {{2,3},{1,2,3}}
%e A320464   7423: {{1,2},{2,3,4}}
%e A320464   7943: {{1,2},{1,2},{2,3}}
%e A320464   8249: {{2,4},{1,2,3}}
%e A320464   8507: {{2,3},{1,2,4}}
%t A320464 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A320464 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A320464 Select[Range[10000],And[normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],PrimeOmega[#]>1]&/@primeMS[#])]&]
%Y A320464 Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.
%Y A320464 Cf. A320456, A320459, A320463, A320532, A320533.
%K A320464 nonn
%O A320464 1,2
%A A320464 _Gus Wiseman_, Oct 13 2018