A320635 - OEIS

%I #4 Oct 19 2018 09:47:34

%S 13,377,611,1363,16211,17719,26273,27521,44603,56173,58609,83291,

%T 91031,91039,99499,141401,147533,203087,301129,315433,467711,761917,

%U 1183403,1280669,1293487,1917929,2075567,2174159,2220907,2415439,2640131

%N MM-numbers of simple labeled connected graphs 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}}.

%e The sequence of terms together with their multiset multisystems begins:

%e        13: {{1,2}}

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

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

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

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

%e     17719: {{1,2},{1,3},{2,3}}

%e     26273: {{1,2},{1,4},{2,3}}

%e     27521: {{1,2},{1,3},{2,4}}

%e     44603: {{1,2},{2,3},{2,4}}

%e     56173: {{1,2},{1,3},{3,4}}

%e     58609: {{1,3},{1,4},{2,3}}

%e     83291: {{1,2},{1,4},{3,4}}

%e     91031: {{1,3},{1,4},{2,4}}

%e     91039: {{1,2},{2,3},{3,4}}

%e     99499: {{1,3},{2,3},{2,4}}

%e    141401: {{1,2},{2,4},{3,4}}

%e    147533: {{1,4},{2,3},{2,4}}

%e    203087: {{1,3},{2,3},{3,4}}

%e    301129: {{1,4},{2,3},{3,4}}

%e    315433: {{1,3},{2,4},{3,4}}

%e    467711: {{1,4},{2,4},{3,4}}

%e    761917: {{1,2},{1,3},{1,4},{2,3}}

%e   1183403: {{1,2},{1,3},{1,4},{2,4}}

%e   1280669: {{1,2},{1,3},{1,4},{1,5}}

%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 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]]]]]]]]];

%t Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#]),Length[zsm[primeMS[#]]]==1]&]

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

%K nonn

%O 1,1

%A _Gus Wiseman_, Oct 18 2018