A324751 -id:A324751 - cp's OEIS frontend

A324748 Number of strict integer partitions of n containing all prime indices of the parts.

1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 2, 3, 2, 2, 4, 3, 4, 3, 5, 6, 9, 8, 7, 8, 11, 12, 13, 15, 17, 22, 22, 20, 28, 31, 32, 36, 41, 43, 53, 53, 59, 70, 76, 77, 89, 99, 108, 124, 135, 139, 160, 172, 188, 209, 229, 243, 274, 298, 315, 353, 391, 417, 457, 496, 538, 588

Offset: 0

Gus Wiseman, Mar 15 2019

nonn

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.

			The first 15 terms count the following integer partitions.
   1: (1)
   3: (2,1)
   5: (4,1)
   6: (3,2,1)
   7: (4,2,1)
   9: (8,1)
   9: (6,2,1)
  10: (4,3,2,1)
  11: (8,2,1)
  11: (5,3,2,1)
  12: (9,2,1)
  12: (7,4,1)
  12: (6,3,2,1)
  13: (8,4,1)
  13: (6,4,2,1)
  14: (8,3,2,1)
  14: (7,4,2,1)
  15: (12,2,1)
  15: (9,3,2,1)
  15: (8,4,2,1)
  15: (5,4,3,2,1)
An example for n = 6 is (20,18,11,5,3,2,1), with prime indices:
  20: {1,1,3}
  18: {1,2,2}
  11: {5}
   5: {3}
   3: {2}
   2: {1}
   1: {}
All of these prime indices {1,2,3,5} belong to the partition, as required.

The subset version is A324736. The non-strict version is A324753. The Heinz number version is A290822. An infinite version is A324698.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A279861, A290689, A290760, A305713.
Cf. A324697, A324737, A324751.

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

A324763 Number of maximal subsets of {2...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 6, 6, 6, 6, 10, 10, 16, 16, 16, 16, 24, 24, 48, 48, 48, 48, 84, 84, 84, 84, 84, 84, 144, 144, 228, 228, 228, 228, 228, 228, 420, 420, 420, 420, 648, 648, 1080, 1080, 1080, 1080, 1800, 1800, 1800, 1800, 1800, 1800, 3600, 3600, 3600, 3600, 3600

Offset: 1

Gus Wiseman, Mar 17 2019

nonn

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.

			The a(1) = 1 through a(9) = 6 subsets:
  {}  {2}  {2}  {2,4}  {3,4}    {2,4,5}  {2,4,5}  {2,4,5,8}  {2,4,5,8}
           {3}  {3,4}  {2,4,5}  {3,4,6}  {2,5,7}  {2,5,7,8}  {2,5,7,8}
                                {4,5,6}  {3,4,6}  {3,4,6,8}  {3,4,6,8,9}
                                         {3,6,7}  {3,6,7,8}  {3,6,7,8,9}
                                         {4,5,6}  {4,5,6,8}  {4,5,6,8,9}
                                         {5,6,7}  {5,6,7,8}  {5,6,7,8,9}

Andrew Howroyd, Table of n, a(n) for n = 1..100

The non-maximal version is A324742.
The version for subsets of {1...n} is A324741.
An infinite version is A304360.
Cf. A076078, A084422, A085945, A112798, A276625, A306844, A324764.
Cf. A324695, A324743, A324751, A324756, A324758, A324762.

maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[2,n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]],{n,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1, k, pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=my(e=0); forstep(k=#p, 1, -1, if(bittest(b,k), e=bitor(e,p[k]), if(!bittest(e,k) && !bitand(p[k], b), return(0)) )); 1);
((k, b)->if(k>#p, ismax(b), my(f=!bitand(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 26 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019

A358453 Number of transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 37, 83, 190, 444, 1051, 2518, 6090, 14852

Offset: 1

Gus Wiseman, Nov 18 2022

We define an unlabeled ordered rooted tree to be transitive if every branch of a branch of the root already appears farther to the left as a branch of the root. An undirected version is A358454.

			The a(1) = 1 through a(7) = 17 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)
                (o(o))  (o(o)o)  (o(o)oo)   (o(o)ooo)
                        (o(oo))  (o(oo)o)   (o(oo)oo)
                        (oo(o))  (o(ooo))   (o(ooo)o)
                                 (oo(o)o)   (o(oooo))
                                 (oo(oo))   (oo(o)oo)
                                 (ooo(o))   (oo(oo)o)
                                 (o(o)(o))  (oo(ooo))
                                            (ooo(o)o)
                                            (ooo(oo))
                                            (oooo(o))
                                            (o(o)(o)o)
                                            (o(o)(oo))
                                            (o(o)o(o))
                                            (o(oo)(o))
                                            (oo(o)(o))
                                            (o(o)((o)))

The unordered version is A290689, ranked by A290822.
The undirected version is A358454, ranked by A358458.
These trees are ranked by A358457.
A000081 counts rooted trees.
A306844 counts anti-transitive rooted trees.
Cf. A318185, A324695, A324751, A324756, A324758, A324764-A324768, A324838, A324840, A324844, A358456.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Function[t,And@@Table[Complement[t[[k]],Take[t,k]]=={},{k,Length[t]}]]]],{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

Gus Wiseman, Mar 17 2019

nonn

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.

			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)))

Cf. A007097, A000081, A290689, A303431, A304360, A306844, A316502, A318186.

Cf. A324695, A324751, A324756, A324758, A324765, A324767, A324769, A324838, A324841, A324844.

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]

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

Gus Wiseman, Mar 17 2019

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.

			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)))))))

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.

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

Gus Wiseman, Mar 17 2019

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.

			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))))))

Gus Wiseman, The a(11) = 128 fully anti-transitive rooted identity trees.

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

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}]

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

Gus Wiseman, Mar 17 2019

nonn

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.

			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))

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

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]

A358456 Number of recursively bi-anti-transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 7, 17, 47, 117, 321, 895, 2556, 7331, 21435, 63116, 187530

Offset: 1

Gus Wiseman, Nov 18 2022

We define an unlabeled ordered rooted tree to be recursively bi-anti-transitive if there are no two branches of the same node such that one is a branch of the other.

			The a(1) = 1 through a(6) = 17 trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((o))o)   (((o))oo)
                          (((oo)))   (((oo))o)
                          ((o)(o))   (((ooo)))
                          (o((o)))   ((o)(oo))
                          ((((o))))  ((oo)(o))
                                     (o((o))o)
                                     (o((oo)))
                                     (oo((o)))
                                     ((((o)))o)
                                     ((((o))o))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

The unordered version is A324765, ranked by A324766.
The directed version is A358455.
A000108 counts ordered rooted trees, unordered A000081.
A306844 counts anti-transitive rooted trees.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A318185, A324695, A324751, A324756, A324758, A324764, A324767, A324768, A324838, A324840, A324844.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],FreeQ[#,{_,x_,_,{_,x_,_},_}|{_,{_,x_,_},_,x_,_}]&]],{n,10}]

A358454 Number of weakly transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 3, 6, 13, 33, 80, 201, 509, 1330, 3432, 8982, 23559, 62189

Offset: 1

Gus Wiseman, Nov 18 2022

We define an unlabeled ordered rooted tree to be weakly transitive if every branch of a branch of the root is itself a branch of the root.

			The a(1) = 1 through a(6) = 13 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)
                ((o)o)  ((o)oo)  ((o)ooo)
                (o(o))  ((oo)o)  ((oo)oo)
                        (o(o)o)  ((ooo)o)
                        (o(oo))  (o(o)oo)
                        (oo(o))  (o(oo)o)
                                 (o(ooo))
                                 (oo(o)o)
                                 (oo(oo))
                                 (ooo(o))
                                 ((o)(o)o)
                                 ((o)o(o))
                                 (o(o)(o))

The unordered version is A290689, ranked by A290822.
The directed version is A358453.
A000081 counts rooted trees.
A306844 counts anti-transitive rooted trees.
Cf. A318185, A324695, A324751, A324756, A324758, A324764-A324768, A324838, A324840, A324844, A358456.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Complement[Union@@#,#]=={}&]],{n,10}]

A358455 Number of recursively anti-transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 72, 206, 608, 1830, 5612, 17442, 54866, 174252, 558072, 1800098

Offset: 1

Gus Wiseman, Nov 18 2022

We define an unlabeled ordered rooted tree to be recursively anti-transitive if no branch of a branch of a subtree is a branch of the same subtree farther to the left.

			The a(1) = 1 through a(5) = 10 trees:
  o  (o)  (oo)   (ooo)    (oooo)
          ((o))  ((o)o)   ((o)oo)
                 ((oo))   ((oo)o)
                 (((o)))  ((ooo))
                          (((o))o)
                          (((o)o))
                          (((oo)))
                          ((o)(o))
                          (o((o)))
                          ((((o))))

The unordered version is A324765, ranked by A324766.
The undirected version is A358456.
A000108 counts ordered rooted trees, unordered A000081.
A306844 counts anti-transitive rooted trees.
A358453 counts transitive ordered trees, unordered A290689.
Cf. A318185, A324695, A324751, A324756, A324758, A324764, A324767, A324768, A324838, A324840, A324844.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],FreeQ[#,{_,x_,_,{_,x_,_},_}]&]],{n,10}]

cp's OEIS Frontend

A324748 Number of strict integer partitions of n containing all prime indices of the parts.

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A324763 Number of maximal subsets of {2...n} containing no prime indices of the elements.

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A358453 Number of transitive ordered rooted trees with n nodes.

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

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

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

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

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A358456 Number of recursively bi-anti-transitive ordered rooted trees with n nodes.

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A358454 Number of weakly transitive ordered rooted trees with n nodes.

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