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.

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A316501 Number of unlabeled rooted trees with n nodes in which the branches of any node with more than one distinct branch have empty intersection.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 45, 103, 250, 611, 1528, 3853, 9875, 25481, 66382, 174085, 459541, 1219462
Offset: 1

Views

Author

Gus Wiseman, Jul 05 2018

Keywords

Examples

			The a(6) = 19 rooted trees:
  (((((o)))))
  ((((oo))))
  (((o(o))))
  (((ooo)))
  ((o((o))))
  ((o(oo)))
  (((o)(o)))
  ((oo(o)))
  ((oooo))
  (o(((o))))
  (o((oo)))
  (o(o(o)))
  (o(ooo))
  ((o)((o)))
  (oo((o)))
  (oo(oo))
  (o(o)(o))
  (ooo(o))
  (ooooo)

Crossrefs

Cf. A000081, A007562, A289509, A302796, A316470, A316473, A316475, A316500, A316502.

Programs

  • Mathematica
    strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],Or[Length[Union[#]]==1,Intersection@@#=={}]&]];
Table[Length[strut[n]],{n,15}]

A316469 Matula-Goebel numbers of unlabeled rooted identity 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, 6, 26, 78, 202, 606, 794, 2382, 2462, 2626, 7386, 7878, 8914, 10322, 12178, 26742, 30966, 32006, 36534, 42374, 43954, 47206, 80194, 96018, 115882, 127122, 131862, 141618, 149782, 158314, 160978, 184622, 217058, 240582, 248662, 260422, 347646, 449346
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 distinct, relatively prime, and already belong to the sequence.

Examples

			78 = prime(1)*prime(2)*prime(6) belongs to the sequence because the indices {1,2,6} are relatively prime, distinct, and already belong to the sequence.
The sequence of all identity RPMG-trees preceded by their Matula-Goebel numbers begins:
     1: o
     2: (o)
     6: (o(o))
    26: (o(o(o)))
    78: (o(o)(o(o)))
   202: (o(o(o(o))))
   606: (o(o)(o(o(o))))
   794: (o(o(o)(o(o))))
  2382: (o(o)(o(o)(o(o))))
  2462: (o(o(o(o(o)))))
  2626: (o(o(o))(o(o(o))))
  7386: (o(o)(o(o(o(o)))))
  7878: (o(o)(o(o))(o(o(o))))

Crossrefs

Cf. A000081, A000837, A004111, A007097, A078374, A276625, A289509, A302696, A302796, A316467, A316470, A316471, A316474, A316494.

Programs

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

A316503 Matula-Goebel numbers 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, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 41, 47, 55, 58, 62, 66, 78, 79, 82, 93, 94, 101, 109, 110, 113, 123, 127, 130, 137, 141, 143, 145, 155, 158, 165, 174, 179, 186, 195, 202, 205, 211, 218, 226, 246, 254, 257, 271, 274, 282, 286, 290, 293
Offset: 1

Views

Author

Gus Wiseman, Jul 05 2018

Keywords

Examples

			Sequence of rooted identity trees preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   6: (o(o))
  10: (o((o)))
  11: ((((o))))
  13: ((o(o)))
  15: ((o)((o)))
  22: (o(((o))))
  26: (o(o(o)))
  29: ((o((o))))
  30: (o(o)((o)))
  31: (((((o)))))

Crossrefs

Cf. A000081, A000837, A004111, A007097, A078374, A276625, A289509, A302796, A316467, A316470, A316494, A316502.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,And[SquareFreeQ[#],Or[PrimeQ[#],GCD@@primeMS[#]==1],And@@#0/@primeMS[#]]]&]

A319379 Number of plane trees with n nodes where the sequence of branches directly under any given node is a chain of multisets.

Original entry on oeis.org

1, 1, 2, 4, 9, 19, 43, 93, 207, 452, 997, 2176, 4776, 10418, 22781, 49674, 108421
Offset: 1

Views

Author

Gus Wiseman, Sep 17 2018

Keywords

Examples

			The a(6) = 19 chain trees:
  (((((o)))))  ((((oo))))  (((ooo)))  ((oooo))  (ooooo)
               (((o)(o)))  ((o)(oo))  (o(ooo))
               (((o(o))))  ((o(oo)))  (oo(oo))
               ((o((o))))  ((oo(o)))  (ooo(o))
               (o(((o))))  (o((oo)))
                           (o(o)(o))
                           (o(o(o)))
                           (oo((o)))

Links

Crossrefs

Cf. A000081, A000108, A001003, A005043, A007562, A118376, A143363, A316470, A319122, A319378, A319380, A319381.

Programs

  • Mathematica
    submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
chnplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[chnplane/@c],And@@submultisetQ@@@Partition[#,2,1]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[chnplane[n]],{n,10}]

A319380 Number of plane trees with n nodes where the sequence of branches directly under any given node is a chain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 17, 30, 53, 94, 169, 303, 543, 968, 1728, 3080, 5491, 9776, 17415, 31008
Offset: 1

Views

Author

Gus Wiseman, Sep 17 2018

Keywords

Examples

			The a(8) = 17 locally identity chain 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((((o))))))  (((o)(o(o))))
                   (o(((((o))))))  (o(((o(o)))))
                                   (o((o((o)))))
                                   (o(o(((o)))))
                                   ((o)(o((o))))
                                   (((o))(o(o)))

Links

Crossrefs

Cf. A000081, A000108, A001003, A005043, A007562, A118376, A316470, A319122, A319379, A319381.

Programs

  • Mathematica
    submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
idchnplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[idchnplane/@c],And[UnsameQ@@#,And@@submultisetQ@@@Partition[#,2,1]]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[idchnplane[n]],{n,10}]

A319381 Number of plane trees with n nodes where the sequence of branches directly under any given node is a membership-chain.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 9, 11, 20, 28, 40, 58, 82, 110, 159, 217, 305, 420, 570, 767, 1042
Offset: 1

Views

Author

Gus Wiseman, Sep 17 2018

Keywords

Examples

			The a(9) = 11 membership-chain 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))(o)o)o)
                                       ((((o))(o))(o))

Links

Crossrefs

Cf. A000081, A000108, A001003, A005043, A007562, A118376, A316470, A319122, A319379, A319380, A319436.

Programs

  • Mathematica
    yanplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[yanplane/@c],And@@MemberQ@@@Partition[#,2,1]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[yanplane[n]],{n,10}]

A319378 Number of plane trees with n nodes where the sequence of branches directly under any given node with at least two branches has empty intersection.

Original entry on oeis.org

1, 1, 2, 5, 13, 39, 118, 375, 1225, 4079, 13794, 47287, 163962, 573717, 2023800
Offset: 1

Views

Author

Gus Wiseman, Sep 17 2018

Keywords

Examples

			The a(5) = 13 locally nonintersecting plane trees:
  ((((o))))  (((oo)))  ((ooo))  (oooo)
             (((o)o))  ((oo)o)
             ((o(o)))  (o(oo))
             (((o))o)  ((o)oo)
             (o((o)))  (o(o)o)
                       (oo(o))

Crossrefs

Cf. A000081, A000108, A001003, A005043, A007562, A118376, A143363, A316470, A319122, A319379, A319380, A319381, A319436.

Programs

  • Mathematica
    monplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[monplane/@c],Or[Length[#]==1,Intersection@@#=={}]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[monplane[n]],{n,10}]

A319271 Number of series-reduced locally non-intersecting aperiodic rooted trees with n nodes.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 3, 9, 12, 27, 42, 91, 151, 312, 550, 1099, 2026, 3999, 7527, 14804, 28336, 55641, 107737, 211851, 413508, 814971, 1600512, 3162761, 6241234
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, and aperiodic if the multiplicities in the multiset of branches directly under any given node are relatively prime, and locally non-intersecting if the branches directly under any given node with more than one branch have empty intersection.

Examples

			The a(8) = 9 rooted trees:
  (o(o(o(o))))
  (o(o(o)(o)))
  (o(ooo(o)))
  (oo(oo(o)))
  (o(o)(o(o)))
  (ooo(o(o)))
  (o(o)(o)(o))
  (ooo(o)(o))
  (ooooo(o))

Crossrefs

Cf. A000081, A000837, A007562, A289509, A301700, A303431, A316470, A316473, A316475, A316495, A319270.

Programs

  • Mathematica
    btrut[n_]:=btrut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[btrut/@c]]]/@IntegerPartitions[n-1],And[Intersection@@#=={},GCD@@Length/@Split[#]==1]&]];
Table[Length[btrut[n]],{n,30}]

A317786 Matula-Goebel numbers of locally connected rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 9, 11, 23, 25, 27, 31, 81, 83, 97, 103, 115, 121, 125, 127, 243, 419, 431, 509, 515, 529, 563, 575, 625, 631, 661, 691, 709, 729, 961, 1067, 1331, 1543, 2095, 2187, 2369, 2575, 2645, 2875, 2897, 3001, 3125, 3637, 3691, 3803, 4091, 4201, 4637, 4663
Offset: 1

Views

Author

Gus Wiseman, Aug 07 2018

Keywords

Comments

An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.

Examples

			The sequence of locally connected trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   9: ((o)(o))
  11: ((((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  81: ((o)(o)(o)(o))
  83: ((((o)(o))))
  97: ((((o))((o))))

Crossrefs

A subset of A184155.
Cf. A000081, A276625, A286518, A286520, A304714, A316470, A316495, A316502, A317077, A317078, A317785.

Programs

  • Mathematica
    multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[OrderedQ[#], UnsameQ@@#, Length[Intersection@@s[[#]]]>0]&]}, If[c=={}, s, csm[Union[Append[Delete[s, List/@c[[1]]], multijoin@@s[[c[[1]]]]]]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[Length[csm[primeMS/@primeMS[n]]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[1000],rupQ[#]&]

A317789 Matula-Goebel numbers of rooted trees that are not locally nonintersecting.

Original entry on oeis.org

9, 21, 23, 25, 27, 39, 46, 49, 57, 63, 65, 69, 73, 81, 83, 87, 91, 92, 97, 103, 111, 115, 117, 121, 125, 129, 133, 138, 146, 147, 159, 161, 166, 167, 169, 171, 183, 184, 185, 189, 194, 199, 203, 206, 207, 213, 219, 227, 230, 235, 237, 243, 247, 249, 253, 259
Offset: 1

Views

Author

Gus Wiseman, Aug 07 2018

Keywords

Comments

An unlabeled rooted tree is locally nonintersecting if there is no common subbranch to all branches directly under any given node.

Examples

			The sequence of rooted trees that are not locally nonintersecting together with their Matula-Goebel numbers begins:
   9: ((o)(o))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  39: ((o)(o(o)))
  46: (o((o)(o)))
  49: ((oo)(oo))
  57: ((o)(ooo))
  63: ((o)(o)(oo))
  65: (((o))(o(o)))
  69: ((o)((o)(o)))
  73: (((o)(oo)))
  81: ((o)(o)(o)(o))
  83: ((((o)(o))))
  87: ((o)(o((o))))
  91: ((oo)(o(o)))
  92: (oo((o)(o)))
  97: ((((o))((o))))

Crossrefs

Cf. A000081, A184155, A276625, A316470, A316495, A316502, A317785, A317786, A317787.

Programs

  • Mathematica
    rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[GCD@@PrimePi/@FactorInteger[n][[All,1]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[100],!rupQ[#]&]
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