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 1-9 of 9 results.

A316471 Number of locally disjoint rooted identity trees with n nodes, meaning no branch overlaps any other branch of the same root.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 21, 43, 89, 185, 391, 840, 1822, 3975, 8727, 19280, 42841, 95661, 214490
Offset: 1

Views

Author

Gus Wiseman, Jul 04 2018

Keywords

Examples

			The a(7) = 11 locally disjoint 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)))

Links

Crossrefs

Cf. A000081, A004111, A276625, A277098, A302696, A303362, A304713, A316467, A316471, A316473, A316474, A316494.

Programs

  • Mathematica
    strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@#&&Select[Tuples[#,2],UnsameQ@@#&&(Intersection@@#=!={})&]=={}&]];
Table[Length[strut[n]],{n,20}]

A306202 Matula-Goebel numbers of rooted semi-identity trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 70, 71, 73, 74, 76, 77, 78, 79, 80, 82, 84, 85
Offset: 1

Views

Author

Gus Wiseman, Jan 29 2019

Keywords

Comments

Definition: A positive integer belongs to the sequence iff its prime indices greater than 1 are distinct and already belong to the sequence. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The sequence of all unlabeled rooted semi-identity trees together with 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)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  30: (o(o)((o)))

Links

Crossrefs

Cf. A000081, A004111, A007097, A276625, A277098, A306200, A306201, A316467.

Programs

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

A316468 Matula-Goebel numbers of locally stable rooted trees, meaning no branch is a submultiset of any other branch of the same root.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 15, 16, 17, 19, 23, 25, 27, 31, 32, 33, 35, 45, 47, 49, 51, 53, 55, 59, 64, 67, 69, 75, 77, 81, 83, 85, 93, 95, 97, 99, 103, 119, 121, 125, 127, 128, 131, 135, 137, 141, 149, 153, 155, 161, 165, 175, 177, 187, 197, 201, 207, 209
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 its distinct prime indices are pairwise indivisible and already belong to the sequence.

Examples

			Sequence of locally stable rooted trees preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  15: ((o)((o)))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))

Crossrefs

Cf. A000081, A004111, A007097, A112798, A277098, A285572, A285573, A303362, A304713, A316467, A316470, A316473, A316475, A316476, A316495.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,And[Select[Tuples[primeMS[#],2],UnsameQ@@#&&Divisible@@#&]=={},And@@#0/@primeMS[#]]]&]

A316474 Number of locally stable rooted identity trees with n nodes, meaning no branch is a subset of any other branch of the same root.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 5, 8, 14, 23, 42, 73, 133, 241, 442, 812, 1508, 2802, 5247, 9842, 18554, 35045, 66453, 126249
Offset: 1

Views

Author

Gus Wiseman, Jul 04 2018

Keywords

Examples

			The a(9) = 8 locally stable rooted identity trees:
((((((((o))))))))
(((((o)((o))))))
((((o)(((o))))))
(((o)((((o))))))
((((o))(((o)))))
((o)(((((o))))))
((o)((o)((o))))
(((o))((((o)))))

Links

Crossrefs

Cf. A000081, A004111, A276625, A277098, A285572, A285573, A303362, A304713, A316467, A316471, A316471, A316475.

Programs

  • Mathematica
    strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@#&&Select[Tuples[#,2],UnsameQ@@#&&Complement@@#=={}&]=={}&]];
Table[Length[strut[n]],{n,20}]

A316494 Matula-Goebel numbers of locally disjoint rooted identity trees, meaning no branch overlaps any other branch of the same root.

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, 79, 82, 93, 94, 101, 109, 110, 113, 123, 127, 137, 141, 143, 145, 155, 158, 165, 179, 186, 202, 205, 211, 218, 226, 246, 254, 257, 271, 274, 282, 286, 290, 293, 310, 317, 327, 330
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 either it is equal to 1, it is a prime number whose prime index already belongs to the sequence, or its prime indices are pairwise coprime, distinct, and already belong to the sequence.

Examples

			The sequence of all locally disjoint 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, A004111, A007097, A276625, A277098, A302696, A303362, A304713, A316467, A316471, A316474, A316495.

Programs

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

A306203 Matula-Goebel numbers of balanced rooted semi-identity trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 11, 16, 17, 19, 21, 31, 32, 53, 57, 59, 64, 67, 73, 85, 127, 128, 131, 133, 159, 241, 256, 269, 277, 311, 331, 335, 365, 367, 371, 393, 399, 439, 512, 649, 709, 719, 739, 751, 917, 933, 937, 1007, 1024, 1113, 1139, 1205, 1241, 1345, 1523
Offset: 1

Views

Author

Gus Wiseman, Jan 29 2019

Keywords

Comments

A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. It is balanced if all leaves are the same distance from the root. The only balanced rooted identity trees are rooted paths.

Examples

			The sequence of all unlabeled balanced rooted semi-identity trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  21: ((o)(oo))
  31: (((((o)))))
  32: (ooooo)
  53: ((oooo))
  57: ((o)(ooo))
  59: ((((oo))))
  64: (oooooo)
  67: (((ooo)))
  73: (((o)(oo)))
  85: (((o))((oo)))

Crossrefs

Cf. A000081, A004111, A007097, A048816, A184155, A276625, A306200, A306201, A306202, A316467, A317710.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
psidQ[n_]:=And[UnsameQ@@DeleteCases[primeMS[n],1],And@@psidQ/@primeMS[n]];
mgtree[n_]:=If[n==1,{},mgtree/@primeMS[n]];
Select[Range[100],And[psidQ[#],SameQ@@Length/@Position[mgtree[#],{}]]&]

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[#]]]&]

A316766 Number of series-reduced locally stable rooted identity trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 30, 72, 180, 458, 1194, 3160, 8459, 22881, 62417, 171526, 474405, 1319395, 3687711, 10352696, 29178988
Offset: 1

Views

Author

Gus Wiseman, Jul 12 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. It is an identity tree if no branch appears multiple times under the same root.

Examples

			The a(6) = 13 trees:
6,
(15),
(1(14)),
(1(1(13))),
(1(1(1(12)))),
(1(23)), (2(13)), (3(12)), (123),
(1(2(12))), (2(1(12))), (12(12)),
(24).
Example of non-stable trees are ((12)(123)) and ((12)(12(12))).

Crossrefs

Cf. A000081, A000669, A001678, A004111, A141268, A292504, A300660, A316467, A316474, A316653, A316654, A316656.

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

Extensions

a(18)-a(21) from Robert Price, Sep 14 2018
Showing 1-9 of 9 results.

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