cp's OEIS Frontend

A309356 MM-numbers of labeled simple covering graphs.

%I A309356 #11 Jul 25 2019 12:25:39
%S A309356 1,13,29,43,47,73,79,101,137,139,149,163,167,199,233,257,269,271,293,
%T A309356 313,347,373,377,389,421,439,443,449,467,487,491,499,559,577,607,611,
%U A309356 631,647,653,673,677,727,751,757,811,821,823,829,839,907,929,937,947,949
%N A309356 MM-numbers of labeled simple covering graphs.
%C A309356 First differs from A322551 in having 377.
%C A309356 Also products of distinct elements of A322551.
%C A309356 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 A309356 Covering means there are no isolated vertices, i.e., the vertex set is the union of the edge set.
%e A309356 The sequence of edge sets together with their MM-numbers begins:
%e A309356     1: {}
%e A309356    13: {{1,2}}
%e A309356    29: {{1,3}}
%e A309356    43: {{1,4}}
%e A309356    47: {{2,3}}
%e A309356    73: {{2,4}}
%e A309356    79: {{1,5}}
%e A309356   101: {{1,6}}
%e A309356   137: {{2,5}}
%e A309356   139: {{1,7}}
%e A309356   149: {{3,4}}
%e A309356   163: {{1,8}}
%e A309356   167: {{2,6}}
%e A309356   199: {{1,9}}
%e A309356   233: {{2,7}}
%e A309356   257: {{3,5}}
%e A309356   269: {{2,8}}
%e A309356   271: {{1,10}}
%e A309356   293: {{1,11}}
%e A309356   313: {{3,6}}
%e A309356   347: {{2,9}}
%e A309356   373: {{1,12}}
%e A309356   377: {{1,2},{1,3}}
%e A309356   389: {{4,5}}
%e A309356   421: {{1,13}}
%t A309356 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A309356 Select[Range[1000],And[SquareFreeQ[#],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]
%Y A309356 Simple graphs are A006125.
%Y A309356 The case for BII-numbers is A326788.
%Y A309356 Cf. A001222, A001358, A006129, A056239, A112798, A302242, A320458, A322551.
%K A309356 nonn
%O A309356 1,2
%A A309356 _Gus Wiseman_, Jul 25 2019