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 1-3 of 3 results.

A324840 Number of fully recursively anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 14, 23, 46, 85, 165, 313, 625, 1225, 2459, 4919, 9928, 20078, 40926, 83592
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is fully recursively anti-transitive if no proper terminal subtree of any terminal subtree is a branch of the larger subtree.

Examples

			The a(1) = 1 through a(7) = 14 fully recursively anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
                 (((o)))  (((oo)))   (((ooo)))    (((oooo)))
                          ((o)(o))   ((o)(oo))    ((o)(ooo))
                          ((((o))))  ((((oo))))   ((oo)(oo))
                                     (((o)(o)))   ((((ooo))))
                                     (((((o)))))  (((o))(oo))
                                                  (((o)(oo)))
                                                  ((o)((oo)))
                                                  ((o)(o)(o))
                                                  (((((oo)))))
                                                  ((((o)(o))))
                                                  (((o))((o)))
                                                  ((((((o))))))

Links

Crossrefs

Cf. A000081, A279861, A290689, A304360, A306844, A318185.
Cf. A324695, A324751, A324756, A324758, A324763, A324765, A324768, A324769, A324770.
Cf. A324838, A324841, A324844, A324846.

Programs

  • Mathematica
    dallt[n_]:=Select[Union[Sort/@Join@@(Tuples[dallt/@#]&/@IntegerPartitions[n-1])],Intersection[Union@@Rest[FixedPointList[Union@@#&,#]],#]=={}&];
Table[Length[dallt[n]],{n,10}]

A324766 Matula-Goebel numbers of recursively anti-transitive rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 40, 44, 46, 49, 50, 51, 53, 57, 59, 62, 63, 64, 67, 68, 71, 73, 77, 79, 80, 81, 83, 85, 87, 88, 92, 93, 95, 97, 99, 100, 103, 109, 115, 118, 121, 124, 125, 127, 128
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

The complement is {6, 12, 13, 14, 15, 18, 24, 26, 28, 30, 36, ...}.
An unlabeled rooted tree is recursively anti-transitive if no branch of a branch of a terminal subtree is a branch of the same subtree.

Examples

			The sequence of recursively anti-transitive rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  10: (o((o)))
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  29: ((o((o))))
  31: (((((o)))))
  32: (ooooo)
  33: ((o)(((o))))
  34: (o((oo)))
  35: (((o))(oo))
  40: (ooo((o)))
  44: (oo(((o))))
  46: (o((o)(o)))
  49: ((oo)(oo))
  50: (o((o))((o)))

Crossrefs

Cf. A007097, A000081, A290689, A303431, A304360, A306844, A316502, A318186.
Cf. A324695, A324751, A324756, A324758, A324765, A324767, A324769, A324838, A324841, A324844.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totantiQ[n_]:=And[Intersection[Union@@primeMS/@primeMS[n],primeMS[n]]=={},And@@totantiQ/@primeMS[n]];
Select[Range[100],totantiQ]

A324769 Matula-Goebel numbers of fully anti-transitive rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 21, 23, 25, 27, 29, 31, 32, 35, 37, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 64, 65, 67, 71, 73, 77, 79, 81, 83, 85, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 129, 131, 133, 137, 139, 143, 147
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is fully anti-transitive if no proper terminal subtree of any branch of the root is a branch of the root.

Examples

			The sequence of fully anti-transitive rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  13: ((o(o)))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  29: ((o((o))))
  31: (((((o)))))
  32: (ooooo)
  35: (((o))(oo))
  37: ((oo(o)))
  41: (((o(o))))
  43: ((o(oo)))
  47: (((o)((o))))
  49: ((oo)(oo))

Crossrefs

Cf. A000081, A007097, A276625, A290760, A304360, A306844.
Cf. A324695, A324751, A324756, A324758, A324766, A324768, A324770.
Cf. A324838, A324841, A324845, A324846.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fullantiQ[n_]:=Intersection[Union@@Rest[FixedPointList[Union@@primeMS/@#&,primeMS[n]]],primeMS[n]]=={};
Select[Range[100],fullantiQ]
Showing 1-3 of 3 results.

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