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%I #5 Oct 15 2018 22:21:50
%S 1,7,13,19,37,53,61,89,91,113,131,133,151,161,223,247,251,259,281,299,
%T 311,329,359,371,377,427,437,463,481,503,593,611,623,659,667,689,703,
%U 719,721,791,793,827,851,863,893,917,923,953,1007,1057,1069,1073,1157
%N MM-numbers of labeled hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.
%C 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 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hypergraph">Hypergraph</a>
%e The sequence of terms together with their multiset multisystems begins:
%e 1: {}
%e 7: {{1,1}}
%e 13: {{1,2}}
%e 19: {{1,1,1}}
%e 37: {{1,1,2}}
%e 53: {{1,1,1,1}}
%e 61: {{1,2,2}}
%e 89: {{1,1,1,2}}
%e 91: {{1,1},{1,2}}
%e 113: {{1,2,3}}
%e 131: {{1,1,1,1,1}}
%e 133: {{1,1},{1,1,1}}
%e 151: {{1,1,2,2}}
%e 161: {{1,1},{2,2}}
%e 223: {{1,1,1,1,2}}
%e 247: {{1,2},{1,1,1}}
%e 251: {{1,2,2,2}}
%e 259: {{1,1},{1,1,2}}
%e 281: {{1,1,2,3}}
%e 299: {{1,2},{2,2}}
%e 311: {{1,1,1,1,1,1}}
%e 329: {{1,1},{2,3}}
%e 359: {{1,1,1,2,2}}
%e 371: {{1,1},{1,1,1,1}}
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
%t Select[Range[1000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[PrimeOmega[#]>1]&/@primeMS[#])]&]
%Y Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.
%Y Cf. A320456, A320461, A320463, A320464, A320533.
%K nonn
%O 1,2
%A _Gus Wiseman_, Oct 14 2018
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