A320635 MM-numbers of simple labeled connected graphs spanning an initial interval of positive integers.

13, 377, 611, 1363, 16211, 17719, 26273, 27521, 44603, 56173, 58609, 83291, 91031, 91039, 99499, 141401, 147533, 203087, 301129, 315433, 467711, 761917, 1183403, 1280669, 1293487, 1917929, 2075567, 2174159, 2220907, 2415439, 2640131

Offset: 1

Author

Gus Wiseman, Oct 18 2018

Keywords

• nonn

Comments

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}}.

Examples

			The sequence of terms together with their multiset multisystems begins:
       13: {{1,2}}
      377: {{1,2},{1,3}}
      611: {{1,2},{2,3}}
     1363: {{1,3},{2,3}}
    16211: {{1,2},{1,3},{1,4}}
    17719: {{1,2},{1,3},{2,3}}
    26273: {{1,2},{1,4},{2,3}}
    27521: {{1,2},{1,3},{2,4}}
    44603: {{1,2},{2,3},{2,4}}
    56173: {{1,2},{1,3},{3,4}}
    58609: {{1,3},{1,4},{2,3}}
    83291: {{1,2},{1,4},{3,4}}
    91031: {{1,3},{1,4},{2,4}}
    91039: {{1,2},{2,3},{3,4}}
    99499: {{1,3},{2,3},{2,4}}
   141401: {{1,2},{2,4},{3,4}}
   147533: {{1,4},{2,3},{2,4}}
   203087: {{1,3},{2,3},{3,4}}
   301129: {{1,4},{2,3},{3,4}}
   315433: {{1,3},{2,4},{3,4}}
   467711: {{1,4},{2,4},{3,4}}
   761917: {{1,2},{1,3},{1,4},{2,3}}
  1183403: {{1,2},{1,3},{1,4},{2,4}}
  1280669: {{1,2},{1,3},{1,4},{1,5}}
		

Crossrefs

Cf. A001222, A007717, A055932, A056239, A112798, A255906, A290103, A302242, A302491, A305078, A320456, A320458.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#]),Length[zsm[primeMS[#]]]==1]&]