A320533 MM-numbers of labeled multi-hypergraphs with multiset edges and no singletons spanning an initial interval of positive integers.

1, 7, 13, 19, 37, 49, 53, 61, 89, 91, 113, 131, 133, 151, 161, 169, 223, 247, 251, 259, 281, 299, 311, 329, 343, 359, 361, 371, 377, 427, 437, 463, 481, 503, 593, 611, 623, 637, 659, 667, 689, 703, 719, 721, 791, 793, 827, 851, 863, 893, 917, 923, 931, 953

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..54.

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}}
   49: {{1,1},{1,1}}
   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}}
  169: {{1,2},{1,2}}
  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}}
  343: {{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[normQ[primeMS/@primeMS[#]], And@@(And[PrimeOmega[#]>1]&/@primeMS[#])]&]

Crossrefs

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

Cf. A320456, A320462, A320463, A320464, A320532.

Sequence in context: A145897 A078863 A263091 * A059351 A320532 A059262

Adjacent sequences: A320530 A320531 A320532 * A320534 A320535 A320536

Keyword

nonn

Author

Gus Wiseman, Oct 14 2018

Status

approved