A309356 - OEIS

%I #11 Jul 25 2019 12:25:39

%S 1,13,29,43,47,73,79,101,137,139,149,163,167,199,233,257,269,271,293,

%T 313,347,373,377,389,421,439,443,449,467,487,491,499,559,577,607,611,

%U 631,647,653,673,677,727,751,757,811,821,823,829,839,907,929,937,947,949

%N MM-numbers of labeled simple covering graphs.

%C First differs from A322551 in having 377.

%C Also products of distinct elements of A322551.

%C A multiset multisystem is a finite multiset of finite multisets. 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}}.

%C Covering means there are no isolated vertices, i.e., the vertex set is the union of the edge set.

%e The sequence of edge sets together with their MM-numbers begins:

%e     1: {}

%e    13: {{1,2}}

%e    29: {{1,3}}

%e    43: {{1,4}}

%e    47: {{2,3}}

%e    73: {{2,4}}

%e    79: {{1,5}}

%e   101: {{1,6}}

%e   137: {{2,5}}

%e   139: {{1,7}}

%e   149: {{3,4}}

%e   163: {{1,8}}

%e   167: {{2,6}}

%e   199: {{1,9}}

%e   233: {{2,7}}

%e   257: {{3,5}}

%e   269: {{2,8}}

%e   271: {{1,10}}

%e   293: {{1,11}}

%e   313: {{3,6}}

%e   347: {{2,9}}

%e   373: {{1,12}}

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

%e   389: {{4,5}}

%e   421: {{1,13}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[1000],And[SquareFreeQ[#],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]

%Y Simple graphs are A006125.

%Y The case for BII-numbers is A326788.

%Y Cf. A001222, A001358, A006129, A056239, A112798, A302242, A320458, A322551.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 25 2019