A319271 -id:A319271 - cp's OEIS frontend

A319270 Numbers that are 1 or whose prime indices are relatively prime and belong to the sequence, and whose prime multiplicities are also relatively prime.

1, 2, 6, 12, 18, 24, 26, 48, 52, 54, 72, 74, 78, 96, 104, 108, 122, 148, 156, 162, 178, 192, 202, 208, 222, 234, 244, 288, 296, 312, 338, 356, 366, 384, 404, 416, 432, 444, 446, 468, 478, 486, 488, 502, 534, 592, 606, 624, 648, 666, 702, 712, 718, 732, 746

Offset: 1

Gus Wiseman, Sep 16 2018

nonn

Also Matula-Goebel numbers of series-reduced locally non-intersecting aperiodic rooted trees.

			The sequence of Matula-Goebel trees of elements of this sequence begins:
   1: o
   2: (o)
   6: (o(o))
  12: (oo(o))
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  48: (oooo(o))
  52: (oo(o(o)))
  54: (o(o)(o)(o))
  72: (ooo(o)(o))
  74: (o(oo(o)))
  78: (o(o)(o(o)))
  96: (ooooo(o))

Cf. A000081, A007097, A061775, A276625, A298748, A301700, A303431, A316470, A319271.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ain[n_]:=Or[n==1,And[GCD@@primeMS[n]==1,GCD@@Length/@Split[primeMS[n]]==1,And@@ain/@primeMS[n]]];
Select[Range[100],ain]

A319292 Number of series-reduced locally nonintersecting rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 1, 6, 48, 455, 5700, 83138, 1454870

Offset: 1

Gus Wiseman, Sep 16 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given node with at least two branches is empty.

			The a(3) = 6 trees are: (1(12)), (112), (1(23)), (2(13)), (3(12)), (123). Missing from this list but counted by A316651 are: (1(11)), (2(11)), (111).

Cf. A000081, A007562, A273873, A289509, A301700, A316470, A316473, A316475, A316651, A319271.

nonintQ[u_]:=Intersection@@u=={};
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],nonintQ]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]

cp's OEIS Frontend

A319270 Numbers that are 1 or whose prime indices are relatively prime and belong to the sequence, and whose prime multiplicities are also relatively prime.

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