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A320463 MM-numbers of labeled simple hypergraphs with no singletons spanning an initial interval of positive integers.

%I A320463 #5 Oct 15 2018 22:21:37
%S A320463 1,13,113,377,611,1291,1363,1469,1937,2021,2117,3277,4537,4859,5249,
%T A320463 5311,7423,8249,8507,16211,16403,16559,16783,16837,17719,20443,20453,
%U A320463 24553,25477,26273,26969,27521,34567,37439,39437,41689,42011,42137,42601,43873,43957
%N A320463 MM-numbers of labeled simple hypergraphs with no singletons spanning an initial interval of positive integers.
%C A320463 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 A320463 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>
%e A320463 The sequence of terms together with their multiset multisystems begins:
%e A320463       1: {}
%e A320463      13: {{1,2}}
%e A320463     113: {{1,2,3}}
%e A320463     377: {{1,2},{1,3}}
%e A320463     611: {{1,2},{2,3}}
%e A320463    1291: {{1,2,3,4}}
%e A320463    1363: {{1,3},{2,3}}
%e A320463    1469: {{1,2},{1,2,3}}
%e A320463    1937: {{1,2},{3,4}}
%e A320463    2021: {{1,4},{2,3}}
%e A320463    2117: {{1,3},{2,4}}
%e A320463    3277: {{1,3},{1,2,3}}
%e A320463    4537: {{1,2},{1,3,4}}
%e A320463    4859: {{1,4},{1,2,3}}
%e A320463    5249: {{1,3},{1,2,4}}
%e A320463    5311: {{2,3},{1,2,3}}
%e A320463    7423: {{1,2},{2,3,4}}
%e A320463    8249: {{2,4},{1,2,3}}
%e A320463    8507: {{2,3},{1,2,4}}
%e A320463   16211: {{1,2},{1,3},{1,4}}
%t A320463 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A320463 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A320463 Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],PrimeOmega[#]>1]&/@primeMS[#])]&]
%Y A320463 Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.
%Y A320463 Cf. A320456, A320458, A320464, A320532, A320533.
%K A320463 nonn
%O A320463 1,2
%A A320463 _Gus Wiseman_, Oct 13 2018