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A320532 MM-numbers of labeled hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers. 8
1, 7, 13, 19, 37, 53, 61, 89, 91, 113, 131, 133, 151, 161, 223, 247, 251, 259, 281, 299, 311, 329, 359, 371, 377, 427, 437, 463, 481, 503, 593, 611, 623, 659, 667, 689, 703, 719, 721, 791, 793, 827, 851, 863, 893, 917, 923, 953, 1007, 1057, 1069, 1073, 1157 (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..53.

Wikipedia, Hypergraph

EXAMPLE

The sequence of terms together with their multiset multisystems begins:

    1: {}

    7: {{1,1}}

   13: {{1,2}}

   19: {{1,1,1}}

   37: {{1,1,2}}

   53: {{1,1,1,1}}

   61: {{1,2,2}}

   89: {{1,1,1,2}}

   91: {{1,1},{1,2}}

  113: {{1,2,3}}

  131: {{1,1,1,1,1}}

  133: {{1,1},{1,1,1}}

  151: {{1,1,2,2}}

  161: {{1,1},{2,2}}

  223: {{1,1,1,1,2}}

  247: {{1,2},{1,1,1}}

  251: {{1,2,2,2}}

  259: {{1,1},{1,1,2}}

  281: {{1,1,2,3}}

  299: {{1,2},{2,2}}

  311: {{1,1,1,1,1,1}}

  329: {{1,1},{2,3}}

  359: {{1,1,1,2,2}}

  371: {{1,1},{1,1,1,1}}

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[1000], And[SquareFreeQ[#], normQ[primeMS/@primeMS[#]], And@@(And[PrimeOmega[#]>1]&/@primeMS[#])]&]

CROSSREFS

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

Cf. A320456, A320461, A320463, A320464, A320533.

Sequence in context: A263091 A320533 A059351 * A059262 A059640 A059643

Adjacent sequences:  A320529 A320530 A320531 * A320533 A320534 A320535

KEYWORD

nonn

AUTHOR

Gus Wiseman, Oct 14 2018

STATUS

approved

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Last modified August 2 07:56 EDT 2021. Contains 346418 sequences. (Running on oeis4.)