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A320464 MM-numbers of labeled multi-hypergraphs with no singletons spanning an initial interval of positive integers. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 21 21:30 EST 2020. Contains 331128 sequences. (Running on oeis4.)