A320464 MM-numbers of labeled multi-hypergraphs with no singletons spanning an initial interval of positive integers.

1, 13, 113, 169, 377, 611, 1291, 1363, 1469, 1937, 2021, 2117, 2197, 3277, 4537, 4859, 4901, 5249, 5311, 7423, 7943, 8249, 8507, 10933, 12769, 16211, 16403, 16559, 16783, 16837, 17719, 19097, 20443, 20453, 24553, 25181, 25477, 26273, 26969, 27521, 28561, 28717

Offset

1,2

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..42.

Wikipedia, Hypergraph

Example

The sequence of terms together with their multiset multisystems begins:
     1: {}
    13: {{1,2}}
   113: {{1,2,3}}
   169: {{1,2},{1,2}}
   377: {{1,2},{1,3}}
   611: {{1,2},{2,3}}
  1291: {{1,2,3,4}}
  1363: {{1,3},{2,3}}
  1469: {{1,2},{1,2,3}}
  1937: {{1,2},{3,4}}
  2021: {{1,4},{2,3}}
  2117: {{1,3},{2,4}}
  2197: {{1,2},{1,2},{1,2}}
  3277: {{1,3},{1,2,3}}
  4537: {{1,2},{1,3,4}}
  4859: {{1,4},{1,2,3}}
  4901: {{1,2},{1,2},{1,3}}
  5249: {{1,3},{1,2,4}}
  5311: {{2,3},{1,2,3}}
  7423: {{1,2},{2,3,4}}
  7943: {{1,2},{1,2},{2,3}}
  8249: {{2,4},{1,2,3}}
  8507: {{2,3},{1,2,4}}

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]];
Select[Range[10000], And[normQ[primeMS/@primeMS[#]], And@@(And[SquareFreeQ[#], PrimeOmega[#]>1]&/@primeMS[#])]&]

Crossrefs

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

Cf. A320456, A320459, A320463, A320532, A320533.

Sequence in context: A044726 A303038 A141937 * A302875 A303523 A244763

Adjacent sequences: A320461 A320462 A320463 * A320465 A320466 A320467

Keyword

nonn

Author

Gus Wiseman, Oct 13 2018

Status

approved