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 1-5 of 5 results.

A317708 Number of aperiodic relatively prime trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 20, 48, 108, 255, 595, 1435, 3434, 8372, 20419, 50289, 124289, 309122, 771508, 1934462
Offset: 1

Views

Author

Gus Wiseman, Aug 05 2018

Keywords

Comments

An unlabeled rooted tree is aperiodic and relatively prime iff either it is a single node or a single node with a single aperiodic relatively prime branch, or the branches directly under any given node have empty intersection (relatively prime) and also have relatively prime multiplicities (aperiodic) and are themselves aperiodic relatively prime trees.

Examples

			The a(6) = 10 aperiodic relatively prime trees:
  (((((o)))))
  (((o(o))))
  ((o((o))))
  ((oo(o)))
  (o(((o))))
  (o(o(o)))
  ((o)((o)))
  (oo((o)))
  (o(o)(o))
  (ooo(o))

Links

Crossrefs

Cf. A000081, A001190, A004111, A301700, A303431, A316501, A316500.
Cf. A317705, A317707, A317709, A317710, A317711, A317712.

Programs

  • Mathematica
    rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,And[Intersection@@#=={},GCD@@Length/@Split[#]==1]]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

A316502 Matula-Goebel numbers of unlabeled rooted 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, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80
Offset: 1

Views

Author

Gus Wiseman, Jul 05 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 either it is a prime or its prime indices are relatively prime, and its prime indices already belong to the sequence.

Examples

			Sequence of rooted trees preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  16: (oooo)

Links

Crossrefs

Cf. A000081, A000837, A007097, A276625, A289509, A302796, A316468, A316470, A316495, A316501, A316502.

Programs

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

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}]

A317787 Number of locally nonintersecting rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 40, 95, 227, 557, 1382, 3485, 8865, 22790, 59022, 153972, 404066, 1066236, 2826885, 7527411, 20121154
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 a(6) = 18 locally nonintersecting rooted trees:
  (((((o)))))
  ((((oo))))
  (((o(o))))
  ((o((o))))
  (o(((o))))
  ((o)((o)))
  (((ooo)))
  ((o(oo)))
  ((oo(o)))
  (o((oo)))
  (o(o(o)))
  (oo((o)))
  (o(o)(o))
  ((oooo))
  (o(ooo))
  (oo(oo))
  (ooo(o))
  (ooooo)
Missing from this list are (((o)(o))) and ((o)(oo)).

Crossrefs

Cf. A000081, A276625, A301700, A316473, A316475, A316501, A316502, A317708, A317785, A317789.

Programs

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

Extensions

a(16)-a(21) from Robert Price, Sep 16 2018

A316772 Number of series-reduced locally nonintersecting rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 11, 27, 75, 202, 565, 1602, 4617, 13472, 39781, 118604, 356605, 1080178, 3293109, 10097356, 31118507, 96341035
Offset: 1

Views

Author

Gus Wiseman, Jul 12 2018

Keywords

Comments

A rooted tree is series-reduced if all non-leaf nodes have at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given root is empty.

Examples

			The a(6) = 27 trees:
6,
(15),
(24),
(1(14)), (114),
(1(23)), (2(13)), (3(12)), (123),
(1(1(13))), (1(113)), (11(13)), (1113),
(1(2(12))), (1(122)), (2(1(12))), (2(112)), (12(12)), (1122),
(1(1(1(12)))), (1(1(112))), (1(11(12))), (1(1112)), (11(1(12))), (11(112)), (111(12)), (11112).

Crossrefs

Cf. A000081, A000669, A001678, A141268, A292504, A316500, A316501, A316503, A316651, A316652, A316655.

Programs

  • Mathematica
    nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],Intersection@@#=={}&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

Extensions

a(17)-a(20) from Robert Price, Sep 14 2018
Showing 1-5 of 5 results.

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