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A320533 MM-numbers of labeled multi-hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.

%I A320533 #4 Oct 15 2018 22:21:57
%S A320533 1,7,13,19,37,49,53,61,89,91,113,131,133,151,161,169,223,247,251,259,
%T A320533 281,299,311,329,343,359,361,371,377,427,437,463,481,503,593,611,623,
%U A320533 637,659,667,689,703,719,721,791,793,827,851,863,893,917,923,931,953
%N A320533 MM-numbers of labeled multi-hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.
%C A320533 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 A320533 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>
%e A320533 The sequence of terms together with their multiset multisystems begins:
%e A320533     1: {}
%e A320533     7: {{1,1}}
%e A320533    13: {{1,2}}
%e A320533    19: {{1,1,1}}
%e A320533    37: {{1,1,2}}
%e A320533    49: {{1,1},{1,1}}
%e A320533    53: {{1,1,1,1}}
%e A320533    61: {{1,2,2}}
%e A320533    89: {{1,1,1,2}}
%e A320533    91: {{1,1},{1,2}}
%e A320533   113: {{1,2,3}}
%e A320533   131: {{1,1,1,1,1}}
%e A320533   133: {{1,1},{1,1,1}}
%e A320533   151: {{1,1,2,2}}
%e A320533   161: {{1,1},{2,2}}
%e A320533   169: {{1,2},{1,2}}
%e A320533   223: {{1,1,1,1,2}}
%e A320533   247: {{1,2},{1,1,1}}
%e A320533   251: {{1,2,2,2}}
%e A320533   259: {{1,1},{1,1,2}}
%e A320533   281: {{1,1,2,3}}
%e A320533   299: {{1,2},{2,2}}
%e A320533   311: {{1,1,1,1,1,1}}
%e A320533   329: {{1,1},{2,3}}
%e A320533   343: {{1,1},{1,1},{1,1}}
%t A320533 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A320533 normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t A320533 Select[Range[1000],And[normQ[primeMS/@primeMS[#]],And@@(And[PrimeOmega[#]>1]&/@primeMS[#])]&]
%Y A320533 Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.
%Y A320533 Cf. A320456, A320462, A320463, A320464, A320532.
%K A320533 nonn
%O A320533 1,2
%A A320533 _Gus Wiseman_, Oct 14 2018