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.

Previous Showing 11-20 of 27 results. Next

A318186 Totally transitive numbers. Matula-Goebel numbers of totally transitive rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 18, 24, 28, 32, 36, 38, 42, 48, 54, 56, 64, 72, 76, 78, 84, 96, 98, 106, 108, 112, 114, 126, 128, 144, 152, 156, 162, 168, 192, 196, 212, 216, 222, 224, 228, 234, 252, 256, 262, 266, 288, 294, 304, 312, 318, 324, 336, 342, 366, 378
Offset: 1

Views

Author

Gus Wiseman, Aug 20 2018

Keywords

Comments

A number x is totally transitive if (1) whenever prime(y) divides x it follows that y is totally transitive and (2) if prime(y) divides x and prime(z) divides y then prime(z) also divides x.

Examples

			The sequence of all totally transitive rooted trees together with 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))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  42: (o(o)(oo))
  48: (oooo(o))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  76: (oo(ooo))
  78: (o(o)(o(o)))
  84: (oo(o)(oo))
  96: (ooooo(o))
  98: (o(oo)(oo))

Crossrefs

Cf. A000081, A001678, A004111, A007097, A061775, A276625, A279861, A290689, A290760, A290822, A291636, A318185, A318187.

Programs

  • Mathematica
    subprimes[n_]:=If[n==1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
trmgQ[n_]:=Or[n==1,And[Divisible[n,Times@@subprimes[n]],And@@Cases[FactorInteger[n],{p_,_}:>trmgQ[PrimePi[p]]]]];
Select[Range[100],trmgQ]

A324767 Number of recursively anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 9, 17, 33, 63, 126, 254, 511, 1039, 2124, 4371, 9059, 18839, 39339, 82385, 173111, 364829, 771010, 1633313
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is recursively anti-transitive if no branch of a branch of any terminal subtree is a branch of the same subtree. It is an identity tree if there are no repeated branches directly under a common root.
Also the number of finitary sets with n brackets where, at any level, no element of an element of a set is an element of the same set. For example, the a(8) = 9 finitary sets are (o = {}):
{{{{{{{o}}}}}}}
{{{{o,{{o}}}}}}
{{{o,{{{o}}}}}}
{{o,{{{{o}}}}}}
{{{o},{{{o}}}}}
{o,{{{{{o}}}}}}
{o,{{o,{{o}}}}}
{{o},{{{{o}}}}}
{{o},{o,{{o}}}}
The Matula-Goebel numbers of these trees are given by A324766.

Examples

			The a(4) = 1 through a(8) = 9 recursively anti-transitive 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)))))   (((o)(((o)))))
                                   ((((((o))))))  (((o(((o))))))
                                                  ((o)((((o)))))
                                                  ((o((((o))))))
                                                  (o(((((o))))))
                                                  (((((((o)))))))

Crossrefs

Cf. A000081, A004111, A276625, A279861, A290689, A290760, A304360, A306844, A316500.
Cf. A324695, A324751, A324758, A324764 (non-recursive version), A324765 (non-identity version), A324766, A324770, A324839, A324840, A324844.

Programs

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

A324770 Number of fully anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 13, 27, 58, 128, 286, 640, 1452, 3308, 7594, 17512, 40591, 94449, 220672
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. It is an identity tree if there are no repeated branches directly under the same root.

Examples

			The a(1) = 1 through a(7) = 6 fully anti-transitive 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))))))

Links

Crossrefs

Cf. A000081, A004111, A276625, A279861, A290760, A304360, A306844.
Cf. A324695, A324751, A324763, A324764, A324765, A324767, A324769.
Cf. A324839, A324840, A324843, A324844, A324846.

Programs

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

A297571 Matula-Goebel numbers of fully unbalanced rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 39, 41, 47, 55, 58, 62, 65, 66, 78, 79, 82, 87, 93, 94, 101, 109, 110, 113, 123, 127, 130, 137, 141, 145, 155, 158, 165, 167, 174, 179, 186, 195, 202, 205, 211, 218, 226, 235, 237, 246, 254, 257, 271, 274
Offset: 1

Views

Author

Gus Wiseman, Dec 31 2017

Keywords

Comments

An unlabeled rooted tree is fully unbalanced if either (1) it is a single node, or (2a) every branch has a different number of nodes and (2b) every branch is fully unbalanced also. The number of fully unbalanced trees with n nodes is A032305(n).
The first finitary number (A276625) not in this sequence is 143.

Examples

			Sequence of fully unbalanced trees 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)))))
  33 ((o)(((o))))
  39 ((o)(o(o)))
  41 (((o(o))))
  47 (((o)((o))))

Crossrefs

Cf. A000081, A003238, A004111, A007097, A032305, A061775, A214577, A273873, A276625, A277098, A290689, A290760, A291441, A291442, A291443.

Programs

  • Mathematica
    nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
MGweight[n_]:=If[n===1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>k*MGweight[PrimePi[p]]]]];
imbalQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[UnsameQ@@MGweight/@m,And@@imbalQ/@m]]];
Select[Range[nn],imbalQ]

A298305 Matula-Goebel numbers of rooted trees with strictly thinning limbs.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 24, 27, 28, 32, 36, 42, 48, 52, 54, 56, 63, 64, 72, 78, 81, 84, 92, 96, 98, 104, 108, 112, 117, 126, 128, 138, 144, 147, 152, 156, 162, 168, 182, 184, 189, 192, 196, 207, 208, 216, 224, 228, 234, 243, 252, 256, 273, 276, 288, 294
Offset: 1

Views

Author

Gus Wiseman, Jan 16 2018

Keywords

Comments

An unlabeled rooted tree has strictly thinning limbs if its outdegrees are strictly decreasing from root to leaves.

Examples

			Sequence of trees begins:
1  o
2  (o)
4  (oo)
6  (o(o))
8  (ooo)
9  ((o)(o))
12 (oo(o))
16 (oooo)
18 (o(o)(o))
24 (ooo(o))
27 ((o)(o)(o))
28 (oo(oo))
32 (ooooo)
36 (oo(o)(o))
42 (o(o)(oo))
48 (oooo(o))
52 (oo(o(o)))
54 (o(o)(o)(o))
56 (ooo(oo))
63 ((o)(o)(oo))
64 (oooooo)
72 (ooo(o)(o))
78 (o(o)(o(o)))
81 ((o)(o)(o)(o))
84 (oo(o)(oo))
92 (oo((o)(o)))
96 (ooooo(o))
98 (o(oo)(oo))

Crossrefs

Cf. A000081, A007097, A061775, A111299, A124343, A124346, A214577, A276625, A290760, A291636, A298126, A298120, A298304.

Programs

  • Mathematica
    MGtree[n_]:=If[n===1,{},MGtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
strthinQ[t_]:=And@@Cases[t,b_List:>Length[b]>Max@@Length/@b,{0,Infinity}];
Select[Range[200],strthinQ[MGtree[#]]&]

A298536 Matula-Goebel numbers of rooted trees such that every branch of the root has a different number of leaves.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 122, 123, 127, 129, 131, 133
Offset: 1

Views

Author

Gus Wiseman, Jan 20 2018

Keywords

Examples

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
5  (((o)))
7  ((oo))
11 ((((o))))
13 ((o(o)))
14 (o(oo))
17 (((oo)))
19 ((ooo))
21 ((o)(oo))
23 (((o)(o)))
26 (o(o(o)))
29 ((o((o))))
31 (((((o)))))
34 (o((oo)))
35 (((o))(oo))
37 ((oo(o)))
38 (o(ooo))
39 ((o)(o(o)))
41 (((o(o))))
43 ((o(oo)))
46 (o((o)(o)))
47 (((o)((o))))

Crossrefs

Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290689, A290760, A291442, A298534, A298535.

Programs

  • Mathematica
    nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
leafcount[n_]:=If[n===1,1,With[{m=primeMS[n]},If[Length[m]===1,leafcount[First[m]],Total[leafcount/@m]]]];
Select[Range[nn],UnsameQ@@leafcount/@primeMS[#]&]

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]

A298126 Matula-Goebel numbers of rooted trees in which all outdegrees are even.

Original entry on oeis.org

1, 4, 14, 16, 49, 56, 64, 86, 106, 196, 224, 256, 301, 344, 371, 424, 454, 526, 622, 686, 784, 886, 896, 1024, 1154, 1204, 1376, 1484, 1589, 1696, 1816, 1841, 1849, 2104, 2177, 2279, 2386, 2401, 2488, 2744, 2809, 2846, 3101, 3136, 3238, 3544, 3584, 3986, 4039
Offset: 1

Views

Author

Gus Wiseman, Jan 13 2018

Keywords

Examples

			Sequence of trees begins:
1   o
4   (oo)
14  (o(oo))
16  (oooo)
49  ((oo)(oo))
56  (ooo(oo))
64  (oooooo)
86  (o(o(oo)))
106 (o(oooo))
196 (oo(oo)(oo))
224 (ooooo(oo))
256 (oooooooo)
301 ((oo)(o(oo)))
344 (ooo(o(oo)))
371 ((oo)(oooo))
424 (ooo(oooo))
454 (o((oo)(oo)))
526 (o(ooo(oo)))
622 (o(oooooo))
686 (o(oo)(oo)(oo))
784 (oooo(oo)(oo))
886 (o(o(o(oo))))
896 (ooooooo(oo))

Links

Crossrefs

Cf. A000081, A001190, A007097, A026424, A028260, A061775, A111299, A214577, A245824, A276625, A277098, A290760, A291441, A291442, A291636, A295461, A297571, A298118, A298120.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
etQ[n_]:=Or[n===1,With[{m=primeMS[n]},EvenQ@Length@m&&And@@etQ/@m]];
Select[Range[10000],etQ]

A298303 Matula-Goebel numbers of rooted trees with thinning limbs.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 58, 60, 62, 63, 64, 65, 66, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 91, 92, 93, 94
Offset: 1

Views

Author

Gus Wiseman, Jan 16 2018

Keywords

Comments

An unlabeled rooted tree has thinning limbs if its outdegrees are weakly decreasing from root to leaves.

Crossrefs

Cf. A000081, A007097, A061775, A111299, A124343, A124346, A214577, A276625, A290760, A291636, A298126, A298120, A298304, A298305.

Programs

  • Mathematica
    MGtree[n_]:=If[n===1,{},MGtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
thinQ[t_]:=And@@Cases[t,b_List:>Length[b]>=Max@@Length/@b,{0,Infinity}];
Select[Range[200],thinQ[MGtree[#]]&]

A298534 Matula-Goebel numbers of rooted trees such that every branch of the root has the same number of leaves.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 27, 29, 30, 31, 32, 33, 36, 37, 40, 41, 43, 44, 45, 47, 48, 49, 50, 53, 54, 55, 59, 60, 61, 62, 64, 66, 67, 71, 72, 73, 75, 79, 80, 81, 83, 88, 89, 90, 91, 93, 96, 97, 99, 100
Offset: 1

Views

Author

Gus Wiseman, Jan 20 2018

Keywords

Examples

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
4  (oo)
5  (((o)))
6  (o(o))
7  ((oo))
8  (ooo)
9  ((o)(o))
10 (o((o)))
11 ((((o))))
12 (oo(o))
13 ((o(o)))
15 ((o)((o)))
16 (oooo)
17 (((oo)))
18 (o(o)(o))
19 ((ooo))
20 (oo((o)))
22 (o(((o))))
23 (((o)(o)))
24 (ooo(o))
25 (((o))((o)))
27 ((o)(o)(o))
29 ((o((o))))
30 (o(o)((o)))

Crossrefs

Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290760, A290822, A291442, A298533, A298536.

Programs

  • Mathematica
    nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
leafcount[n_]:=If[n===1,1,With[{m=primeMS[n]},If[Length[m]===1,leafcount[First[m]],Total[leafcount/@m]]]];
Select[Range[nn],SameQ@@leafcount/@primeMS[#]&]
Previous Showing 11-20 of 27 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.