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

1,1

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}}.

Links

Table of n, a(n) for n=1..31.

Example

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}}

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]&]

Crossrefs

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

Sequence in context: A082403 A171656 A320458 * A279306 A134498 A329554

Adjacent sequences: A320632 A320633 A320634 * A320636 A320637 A320638

Keyword

nonn

Author

Gus Wiseman, Oct 18 2018

Status

approved