cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A316470 Matula-Goebel numbers of unlabeled rooted RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 18, 24, 26, 28, 32, 36, 38, 42, 48, 52, 54, 56, 64, 72, 74, 76, 78, 84, 86, 96, 98, 104, 106, 108, 112, 114, 122, 126, 128, 144, 148, 152, 156, 162, 168, 172, 178, 182, 192, 196, 202, 208, 212, 214, 216, 222, 224, 228, 234, 244, 252
Offset: 1

Views

Author

Gus Wiseman, Jul 04 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. A number is in the sequence iff it is 1 or its prime indices are relatively prime and already belong to the sequence.

Examples

			The sequence of all RPMG-trees preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  42: (o(o)(oo))

Crossrefs

Cf. A000081, A000837, A007097, A289509, A302796, A316468, A316469, A316473, A316475, A316495.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Or[#==1,And[GCD@@primeMS[#]==1,And@@#0/@primeMS[#]]]&]

A316467 Matula-Goebel numbers of locally stable rooted identity trees, meaning no branch is a subset of any other branch of the same root.

Original entry on oeis.org

1, 2, 3, 5, 11, 15, 31, 33, 47, 55, 93, 127, 137, 141, 155, 165, 211, 257, 341, 381, 411, 465, 487, 633, 635, 709, 771, 773, 811, 907, 977, 1023, 1055, 1285, 1297, 1397, 1457, 1461, 1507, 1621, 1705, 1905, 2127, 2293, 2319, 2321, 2433, 2621, 2721, 2833, 2931
Offset: 1

Views

Author

Gus Wiseman, Jul 04 2018

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. A number belongs to this sequence iff it is squarefree, its distinct prime indices are pairwise indivisible, and its prime indices also belong to this sequence.

Examples

			165 = prime(2)*prime(3)*prime(5) belongs to the sequence because it is squarefree, the indices {2,3,5} are pairwise indivisible, and each of them already belongs to the sequence.
Sequence of locally stable rooted identity trees preceded by their Matula-Goebel numbers begins:
    1: o
    2: (o)
    3: ((o))
    5: (((o)))
   11: ((((o))))
   15: ((o)((o)))
   31: (((((o)))))
   33: ((o)(((o))))
   47: (((o)((o))))
   55: (((o))(((o))))
   93: ((o)((((o)))))
  127: ((((((o))))))
  137: (((o)(((o)))))
  141: ((o)((o)((o))))
  155: (((o))((((o)))))
  165: ((o)((o))(((o))))

Crossrefs

Cf. A000081, A004111, A007097, A276625, A277098, A285572, A285573, A302796, A303362, A304713, A316468, A316469, A316471, A316474, A316476, A316494.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ain[n_]:=And[Select[Tuples[primeMS[n],2],UnsameQ@@#&&Divisible@@#&]=={},SquareFreeQ[n],And@@ain/@primeMS[n]];
Select[Range[100],ain]

A316500 Number of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 22, 46, 96, 205, 442, 976, 2146, 4789, 10719, 24202, 54841, 124967, 285724, 656011, 1510929, 3491151, 8088692, 18790084
Offset: 1

Views

Author

Gus Wiseman, Jul 05 2018

Keywords

Examples

			The a(7) = 11 rooted identity trees:
  ((((((o))))))
  ((((o(o)))))
  (((o((o)))))
  ((o(((o)))))
  ((o(o(o))))
  (((o)((o))))
  (o((((o)))))
  (o((o(o))))
  (o(o((o))))
  ((o)(((o))))
  (o(o)((o)))

Crossrefs

Cf. A000081, A004111, A007562, A078374, A289509, A316469, A316471, A316474, A316501, A316503.

Programs

  • Mathematica
    strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@#&&Or[Length[#]==1,Intersection@@#=={}]&]];
Table[Length[strut[n]],{n,20}]

A319285 Number of series-reduced locally stable rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 2, 9, 69, 619, 7739, 109855, 1898230
Offset: 1

Views

Author

Gus Wiseman, Sep 16 2018

Keywords

Comments

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally stable if no branch is a submultiset of any other branch of the same root.

Examples

			The a(3) = 9 trees:
  (1(11))
   (111)
  (1(12))
  (2(11))
   (112)
  (1(23))
  (2(13))
  (3(12))
   (123)
Examples of rooted trees that are not locally stable are ((11)(111)), ((11)(112)), ((12)(112)), ((12)(123)).

Crossrefs

Cf. A000081, A316466, A316468, A316469, A316473, A316475.

Programs

  • Mathematica
    submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
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&])],stableQ]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]
Showing 1-4 of 4 results.

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