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 1-10 of 25 results. Next

A290689 Number of transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 88, 143, 229, 370, 592, 955, 1527, 2457, 3929, 6304, 10081
Offset: 1

Views

Author

Gus Wiseman, Oct 19 2017

Keywords

Comments

A rooted tree is transitive if every proper terminal subtree is also a branch of the root. First differs from A206139 at a(13) = 143.
Regarding the notation, a rooted tree is a finite multiset of rooted trees. For example, the rooted tree (o(o)(oo)) is short for {{},{{}},{{},{}}}. Each "o" is a leaf. Each pair of parentheses corresponds to a non-leaf node (such as the root). Its contents "(...)" represent a branch. - Gus Wiseman, Nov 16 2024

Examples

			The a(7) = 8 7-node transitive rooted trees are: (o(oooo)), (oo(ooo)), (o(o)((o))), (o(o)(oo)), (ooo(oo)), (oo(o)(o)), (oooo(o)), (oooooo).

Links

Crossrefs

The restriction to identity trees (A004111) is A279861, ranks A290760.
These trees are ranked by A290822.
The anti-transitive version is A306844, ranks A324758.
The totally transitive case is A318185 (by leaves A318187), ranks A318186.
A version for integer partitions is A324753, for subsets A324736.
The ordered version is A358453, ranks A358457, undirected A358454.
Cf. A000081, A001192, A001678, A324770, A324969, A325705, A358455, A358460.

Programs

  • Mathematica
    nn=18;
rtall[n_]:=If[n===1,{{}},Module[{cas},Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])]]];
Table[Length[Select[rtall[n],Complement[Union@@#,#]==={}&]],{n,nn}]

Extensions

a(20) from Robert Price, Sep 13 2018
a(21)-a(22) from Robert P. P. McKone, Dec 16 2023

A290822 Transitive numbers: Matula-Goebel numbers of transitive rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 18, 24, 28, 30, 32, 36, 38, 42, 48, 54, 56, 60, 64, 72, 76, 78, 84, 90, 96, 98, 106, 108, 112, 114, 120, 126, 128, 138, 144, 150, 152, 156, 162, 168, 180, 192, 196, 210, 212, 216, 222, 224, 228, 234, 238, 240, 252, 256, 262, 266, 270
Offset: 1

Views

Author

Gus Wiseman, Oct 19 2017

Keywords

Comments

A number x is transitive if whenever prime(y) divides x and prime(z) divides y, we have prime(z) divides x.

Examples

			The sequence of transitive rooted trees 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))
30 (o(o)((o)))
32 (ooooo)
36 (oo(o)(o))
38 (o(ooo))
42 (o(o)(oo))
48 (oooo(o))
54 (o(o)(o)(o))
56 (ooo(oo))
60 (oo(o)((o)))
64 (oooooo)
72 (ooo(o)(o))
76 (oo(ooo))
78 (o(o)(o(o)))
84 (oo(o)(oo))
90 (o(o)(o)((o)))
96 (ooooo(o))
98 (o(oo)(oo))

Links

Crossrefs

Cf. A007097, A276625, A279861, A290689, A290760.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
subprimes[n_]:=If[n===1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
Select[Range[270],Divisible[#,Times@@subprimes[#]]&]

A306844 Number of anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 36, 83, 212, 532, 1379, 3577, 9444, 25019, 66943, 179994, 487031, 1323706, 3614622, 9907911
Offset: 1

Views

Author

Gus Wiseman, Mar 13 2019

Keywords

Comments

A rooted tree is anti-transitive if the subbranches are disjoint from the branches, i.e., no branch of a branch is a branch.

Examples

			The a(1) = 1 through a(6) = 14 anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          ((o(o)))   ((o(oo)))
                          (o((o)))   ((oo(o)))
                          ((((o))))  (o((oo)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     (((o(o))))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Links

Crossrefs

Cf. A276625, A279861, A279861, A290689, A290760, A304360.
Cf. A324694, A324695, A324738, A324741, A324743, A324751, A324754, A324756, A324758, A324759, A324764.

Programs

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

Extensions

a(16)-a(20) from Jinyuan Wang, Jun 20 2020

A290760 Matula-Goebel numbers of transitive rooted identity trees (or transitive finitary sets).

Original entry on oeis.org

1, 2, 6, 30, 78, 330, 390, 870, 1410, 3198, 3390, 4290, 7878, 9570, 10230, 11310, 13026, 15510, 15990, 18330, 26070, 30966, 37290, 39390, 40890, 44070, 45210, 65130, 84810, 94830, 98310, 104610, 122070, 124410, 132990, 154830, 159330, 175890, 198330, 201630
Offset: 1

Views

Author

Gus Wiseman, Oct 19 2017

Keywords

Comments

A rooted tree is transitive if every terminal subtree is a branch of the root. A finitary set is transitive if every element is also a subset.

Examples

			Let o = {}. The sequence of transitive finitary sets begins:
1     o
2     {o}
6     {o,{o}}
30    {o,{o},{{o}}}
78    {o,{o},{o,{o}}}
330   {o,{o},{{o}},{{{o}}}}
390   {o,{o},{{o}},{o,{o}}}
870   {o,{o},{{o}},{o,{{o}}}}
1410  {o,{o},{{o}},{{o},{{o}}}}
3198  {o,{o},{o,{o}},{{o,{o}}}}
3390  {o,{o},{{o}},{o,{o},{{o}}}}
4290  {o,{o},{{o}},{{{o}}},{o,{o}}}
7878  {o,{o},{o,{o}},{o,{o,{o}}}}
9570  {o,{o},{{o}},{{{o}}},{o,{{o}}}}
10230 {o,{o},{{o}},{{{o}}},{{{{o}}}}}
11310 {o,{o},{{o}},{o,{o}},{o,{{o}}}}
13026 {o,{o},{o,{o}},{{o},{o,{o}}}}
15510 {o,{o},{{o}},{{{o}}},{{o},{{o}}}}
15990 {o,{o},{{o}},{o,{o}},{{o,{o}}}}
18330 {o,{o},{{o}},{o,{o}},{{o},{{o}}}}

Crossrefs

Cf. A000081, A001192, A004111, A007097, A076146, A276625, A279861, A290689, A290822.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
finitaryQ[n_]:=finitaryQ[n]=Or[n===1,With[{m=primeMS[n]},{UnsameQ@@m,finitaryQ/@m}]/.List->And];
subprimes[n_]:=If[n===1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
transitaryQ[n_]:=Divisible[n,Times@@subprimes[n]];
nn=100000;Fold[Select,Range[nn],{finitaryQ,transitaryQ}]

A318185 Number of totally transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 12, 17, 28, 41, 65, 96, 150, 221, 342, 506, 771, 1142, 1731, 2561, 3855, 5702, 8538, 12620, 18817, 27774, 41276, 60850, 90139
Offset: 1

Views

Author

Gus Wiseman, Aug 20 2018

Keywords

Comments

A rooted tree is totally transitive if every branch of the root is totally transitive and every branch of a branch of the root is also a branch of the root. Unlike transitive rooted trees (A290689), every terminal subtree of a totally transitive rooted tree is itself totally transitive.

Examples

			The a(8) = 12 totally transitive rooted trees:
  (o(o)(o(o)))
  (o(o)(o)(o))
  (o(o)(ooo))
  (o(oo)(oo))
  (oo(o)(oo))
  (ooo(o)(o))
  (o(ooooo))
  (oo(oooo))
  (ooo(ooo))
  (oooo(oo))
  (ooooo(o))
  (ooooooo)
The a(9) = 17 totally transitive rooted trees:
  (o(o)(oo(o)))
  (oo(o)(o(o)))
  (o(o)(o)(oo))
  (oo(o)(o)(o))
  (o(o)(oooo))
  (o(oo)(ooo))
  (oo(o)(ooo))
  (oo(oo)(oo))
  (ooo(o)(oo))
  (oooo(o)(o))
  (o(oooooo))
  (oo(ooooo))
  (ooo(oooo))
  (oooo(ooo))
  (ooooo(oo))
  (oooooo(o))
  (oooooooo)

Crossrefs

Cf. A000081, A001678, A004111, A279861, A290689, A290760, A290822, A318186, A318187.

Programs

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

A324736 Number of subsets of {1...n} containing all prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 4, 7, 9, 15, 22, 43, 79, 127, 175, 343, 511, 851, 1571, 3141, 4397, 8765, 13147, 25243, 46843, 76795, 115171, 230299, 454939, 758203, 1516363, 2916079, 4356079, 8676079, 12132079, 24264157, 45000157, 73800253, 145685053, 291369853, 437054653, 728424421
Offset: 0

Views

Author

Gus Wiseman, Mar 13 2019

Keywords

Comments

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.
Also the number of subsets of {1...n} containing no prime indices of the non-elements up to n.

Examples

			The a(0) = 1 through a(6) = 15 subsets:
  {}  {}   {}     {}       {}         {}           {}
      {1}  {1}    {1}      {1}        {1}          {1}
           {1,2}  {1,2}    {1,2}      {1,2}        {1,2}
                  {1,2,3}  {1,4}      {1,4}        {1,4}
                           {1,2,3}    {1,2,3}      {1,2,3}
                           {1,2,4}    {1,2,4}      {1,2,4}
                           {1,2,3,4}  {1,2,3,4}    {1,2,6}
                                      {1,2,3,5}    {1,2,3,4}
                                      {1,2,3,4,5}  {1,2,3,5}
                                                   {1,2,3,6}
                                                   {1,2,4,6}
                                                   {1,2,3,4,5}
                                                   {1,2,3,4,6}
                                                   {1,2,3,5,6}
                                                   {1,2,3,4,5,6}
An example for n = 18 is {1,2,4,7,8,9,12,16,17,18}, whose elements have the following prime indices:
   1: {}
   2: {1}
   4: {1,1}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  12: {1,1,2}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
All of these prime indices {1,2,4,7} belong to the subset, as required.

Links

Crossrefs

The strict integer partition version is A324748. The integer partition version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A304360, A320426.
Cf. A324697, A324737, A324741, A324743.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,10}]
  • PARI
    pset(n)={my(b=0, f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 15 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 15 2019

A324764 Number of anti-transitive rooted identity trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 3, 4, 9, 20, 41, 89, 196, 443, 987, 2246, 5114, 11757, 27122, 62898, 146392, 342204, 802429, 1887882
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. It is anti-transitive if the branches of the branches of the root are disjoint from the branches of the root.
Also the number of finitary sets S with n brackets where no element of an element of S is also an element of S. For example, the a(8) = 20 finitary sets are (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},{o,{{o}}}}
{{{o}},{o,{o}}}

Examples

			The a(1) = 1 through a(7) = 9 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))))))

Links

Crossrefs

Cf. A000081, A004111, A276625, A279861, A290689, A304360, A306844 (non-identity version), A316500, A317787, A318185.
Cf. A324694, A324751, A324756, A324758, A324765, A324767, A324768, A324770, A324839, A324840, A324844.

Programs

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

Extensions

a(21)-a(22) from Jinyuan Wang, Jun 20 2020

A324741 Number of subsets of {1...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 19, 30, 54, 96, 156, 248, 440, 688, 1120, 1864, 3664, 5856, 11232, 16896, 31296, 53952, 91008, 137472, 270528, 516720, 863088, 1710816, 3173856, 4836672, 9329472, 14897376, 29788128, 52256448, 88429248, 166037184, 331648704, 497685888, 829449600
Offset: 0

Views

Author

Gus Wiseman, Mar 15 2019

Keywords

Comments

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 a(0) = 1 through a(6) = 19 subsets:
  {}  {}   {}   {}     {}     {}       {}
      {1}  {1}  {1}    {1}    {1}      {1}
           {2}  {2}    {2}    {2}      {2}
                {3}    {3}    {3}      {3}
                {1,3}  {4}    {4}      {4}
                       {1,3}  {5}      {5}
                       {2,4}  {1,3}    {6}
                       {3,4}  {1,5}    {1,3}
                              {2,4}    {1,5}
                              {2,5}    {2,4}
                              {3,4}    {2,5}
                              {4,5}    {3,4}
                              {2,4,5}  {3,6}
                                       {4,5}
                                       {4,6}
                                       {5,6}
                                       {2,4,5}
                                       {3,4,6}
                                       {4,5,6}
An example for n = 20 is {5,6,7,9,10,12,14,15,16,19,20}, with prime indices:
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  15: {2,3}
  16: {1,1,1,1}
  19: {8}
  20: {1,1,3}
None of these prime indices {1,2,3,4,8} belong to the subset, as required.

Links

Crossrefs

The maximal case is A324743. The strict integer partition version is A324751. The integer partition version is A324756. The Heinz number version is A324758. An infinite version is A304360.
Cf. A000720, A001462, A007097, A076078, A084422, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324736, A324742.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Range[n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,10}]
  • PARI
    pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,pset(k)), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(!bitand(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

A324753 Number of integer partitions of n containing all prime indices of their parts.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 7, 8, 14, 16, 23, 29, 40, 49, 66, 81, 109, 133, 172, 211, 274, 332, 419, 511, 640, 775, 965, 1165, 1434, 1730, 2109, 2530, 3083, 3683, 4447, 5308, 6375, 7573, 9062, 10730, 12786, 15104, 17909, 21095, 24937, 29284, 34488, 40421, 47450
Offset: 0

Views

Author

Gus Wiseman, Mar 16 2019

Keywords

Comments

These could be described as transitive integer partitions.
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 a(1) = 1 through a(8) = 8 integer partitions:
  (1)  (11)  (21)   (211)   (41)     (321)     (421)      (3221)
             (111)  (1111)  (221)    (411)     (2221)     (4211)
                            (2111)   (2211)    (3211)     (22211)
                            (11111)  (21111)   (4111)     (32111)
                                     (111111)  (22111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (2111111)
                                                          (11111111)

Crossrefs

The subset version is A324736. The strict case is A324748. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A000837, A001462, A007097, A051424, A112798, A276625, A279861, A290689, A290760.
Cf. A324697, A324737.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],SubsetQ[#,PrimePi/@First/@Join@@FactorInteger/@DeleteCases[#,1]]&]],{n,0,30}]

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}]
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