A320532 - OEIS

%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