A309356 MM-numbers of labeled simple covering graphs.

1, 13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 199, 233, 257, 269, 271, 293, 313, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 559, 577, 607, 611, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 907, 929, 937, 947, 949

Offset: 1

Author

Gus Wiseman, Jul 25 2019

Keywords

• nonn

Comments

First differs from A322551 in having 377.

Also products of distinct elements of A322551.

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}}.

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

Examples

			The sequence of edge sets together with their MM-numbers begins:
    1: {}
   13: {{1,2}}
   29: {{1,3}}
   43: {{1,4}}
   47: {{2,3}}
   73: {{2,4}}
   79: {{1,5}}
  101: {{1,6}}
  137: {{2,5}}
  139: {{1,7}}
  149: {{3,4}}
  163: {{1,8}}
  167: {{2,6}}
  199: {{1,9}}
  233: {{2,7}}
  257: {{3,5}}
  269: {{2,8}}
  271: {{1,10}}
  293: {{1,11}}
  313: {{3,6}}
  347: {{2,9}}
  373: {{1,12}}
  377: {{1,2},{1,3}}
  389: {{4,5}}
  421: {{1,13}}
		

Crossrefs

Simple graphs are A006125.

The case for BII-numbers is A326788.

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

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],And[SquareFreeQ[#],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]