A320463 - OEIS

%I #5 Oct 15 2018 22:21:37

%S 1,13,113,377,611,1291,1363,1469,1937,2021,2117,3277,4537,4859,5249,

%T 5311,7423,8249,8507,16211,16403,16559,16783,16837,17719,20443,20453,

%U 24553,25477,26273,26969,27521,34567,37439,39437,41689,42011,42137,42601,43873,43957

%N MM-numbers of labeled simple hypergraphs with 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      13: {{1,2}}

%e     113: {{1,2,3}}

%e     377: {{1,2},{1,3}}

%e     611: {{1,2},{2,3}}

%e    1291: {{1,2,3,4}}

%e    1363: {{1,3},{2,3}}

%e    1469: {{1,2},{1,2,3}}

%e    1937: {{1,2},{3,4}}

%e    2021: {{1,4},{2,3}}

%e    2117: {{1,3},{2,4}}

%e    3277: {{1,3},{1,2,3}}

%e    4537: {{1,2},{1,3,4}}

%e    4859: {{1,4},{1,2,3}}

%e    5249: {{1,3},{1,2,4}}

%e    5311: {{2,3},{1,2,3}}

%e    7423: {{1,2},{2,3,4}}

%e    8249: {{2,4},{1,2,3}}

%e    8507: {{2,3},{1,2,4}}

%e   16211: {{1,2},{1,3},{1,4}}

%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[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],PrimeOmega[#]>1]&/@primeMS[#])]&]

%Y Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

%Y Cf. A320456, A320458, A320464, A320532, A320533.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 13 2018