A324764 -id:A324764 - cp's OEIS frontend

A324750 Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

1, 0, 1, 1, 1, 1, 2, 3, 2, 4, 4, 4, 6, 8, 8, 11, 10, 15, 16, 19, 23, 27, 28, 35, 39, 47, 50, 63, 68, 77, 91, 102, 114, 130, 147, 169, 187, 213, 237, 268, 300, 336, 380, 422, 472, 525, 587, 647, 731, 810, 895, 996, 1102, 1227, 1355, 1498, 1661, 1818, 2020, 2221

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 a(2) = 1 through a(17) = 15 strict integer partitions (A...H = 10...17):
  2  3  4  5  6   7   8   9   A   B    C    D    E    F    G    H
              42  43  62  54  64  65   75   76   86   87   97   98
                  52      63  73  83   84   85   95   96   A6   A7
                          72  82  542  93   94   A4   A5   C4   B6
                                       A2   A3   B3   B4   D3   C5
                                       642  B2   C2   C3   E2   D4
                                            643  752  D2   763  E3
                                            652  842  654  862  F2
                                                      762  943  854
                                                      843  A42  863
                                                      852       872
                                                                A43
                                                                A52
                                                                B42
                                                                6542

The subset version is A324739. The non-strict version is A324755. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A304360, A305713, A306844.
Cf. A324696, A324737, A324742, A324744, A324764.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,30}]

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}]

A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.

Original entry on oeis.org

1, 2, 3, 6, 10, 20, 30, 60, 96, 192, 312, 624, 936, 1872, 3744, 7488, 12480, 24960, 37440, 74880, 142848, 285696, 456192, 912384, 1548288, 3096576, 5308416, 10616832, 15925248, 31850496, 51978240, 103956480, 200835072, 401670144, 771489792, 1542979584, 2314469376

Offset: 1

Gus Wiseman, Mar 14 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(6) = 20 subsets:
  {}  {}   {}   {}     {}       {}
      {2}  {2}  {2}    {2}      {2}
           {3}  {3}    {3}      {3}
                {4}    {4}      {4}
                {2,4}  {5}      {5}
                {3,4}  {2,4}    {6}
                       {2,5}    {2,4}
                       {3,4}    {2,5}
                       {4,5}    {2,6}
                       {2,4,5}  {3,4}
                                {3,6}
                                {4,5}
                                {4,6}
                                {5,6}
                                {2,4,5}
                                {2,4,6}
                                {2,5,6}
                                {3,4,6}
                                {4,5,6}
                                {2,4,5,6}

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

The maximal case is A324762. The case of subsets of {1...n} is A324738. The strict integer partition version is A324750. The integer partition version is A324755. The Heinz number version is A324760. An infinite version is A324694.
Cf. A000720, A001221, A001462, A007097, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324696, A324736, A324737, A324741, A324742, A324744, A324764.

Table[Length[Select[Subsets[Range[2,n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{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,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 16 2019

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

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}]

A324752 Number of strict integer partitions of n not containing 1 or any prime indices of the parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 1, 4, 4, 4, 5, 6, 7, 10, 9, 12, 12, 16, 17, 22, 22, 26, 31, 35, 37, 46, 50, 55, 66, 70, 82, 90, 101, 114, 127, 143, 159, 172, 202, 215, 246, 267, 301, 327, 366, 402, 447, 491, 545, 600, 655, 722, 795, 875, 964, 1050, 1152, 1259, 1383

Offset: 0

Gus Wiseman, Mar 16 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(2) = 1 through a(17) = 12 strict integer partitions (A...H = 10...17):
  2  3  4  5  6   7   8  9   A   B    C   D    E    F    G    H
              42  43     54  64  65   75  76   86   87   97   98
                  52     63  73  83   84  85   95   96   A6   A7
                         72  82  542  93  94   A4   A5   C4   B6
                                      A2  B2   B3   B4   D3   C5
                                          643  752  C3   E2   D4
                                               842  D2   763  E3
                                                    654  943  854
                                                    843  A42  863
                                                    852       872
                                                              A52
                                                              B42
An example for n = 60 is (19,14,13,7,5,2), with prime indices:
  19: {8}
  14: {1,4}
  13: {6}
   7: {4}
   5: {3}
   2: {1}
None of these prime indices {1,3,4,6,8} belong to the partition, as required.

The subset version is A324742, with maximal case is A324763. The non-strict version is A324757. The Heinz number version is A324761. An infinite version is A304360.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713, A306844, A324764.
Cf. A324695, A324743, A324748, A324751, A324756, A324758.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]

A324757 Number of integer partitions of n not containing 1 or any prime indices of the parts.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 4, 3, 4, 6, 9, 7, 14, 12, 19, 21, 28, 29, 41, 45, 56, 64, 81, 89, 114, 125, 154, 176, 211, 236, 288, 324, 383, 432, 514, 578, 678, 766, 891, 1006, 1176, 1306, 1525, 1711, 1966, 2212, 2538, 2839, 3258, 3646, 4150, 4647, 5288, 5891, 6698, 7472

Offset: 0

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(2) = 1 through a(10) = 9 integer partitions:
  (2)  (3)  (4)   (5)  (6)    (7)   (8)     (9)    (A)
            (22)       (33)   (43)  (44)    (54)   (55)
                       (42)   (52)  (422)   (63)   (64)
                       (222)        (2222)  (72)   (73)
                                            (333)  (82)
                                            (522)  (433)
                                                   (442)
                                                   (4222)
                                                   (22222)

The subset version is A324742, with maximal case A324763. The strict case is A324752. The Heinz number version is A324761. An infinite version is A324695.

Cf. A000720, A000837, A001462, A051424, A112798, A276625, A290822, A304360, A306844, A324764, A324742, A324753, A324756.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]

A324761 Heinz numbers of integer partitions not containing 1 or any prime indices of the parts.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 113, 115, 121, 123, 125, 127, 129, 131, 133, 137, 139, 143, 147, 149

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 Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  41: {13}
  43: {14}

The subset version is A324742, with maximal case A324763. The strict integer partition version is A324752. The integer partition version is A324757. An infinite version is A324695.

Cf. A000720, A001221, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324743, A324751, A324756, A324758, A324764.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],Intersection[primeMS[#],Union@@primeMS/@primeMS[#]]=={}&]

cp's OEIS Frontend

A324750 Number of strict integer partitions of n not containing 1 or any part whose prime indices all belong to the partition.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A324739 Number of subsets of {2...n} containing no element whose prime indices all belong to the subset.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A324752 Number of strict integer partitions of n not containing 1 or any prime indices of the parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A324757 Number of integer partitions of n not containing 1 or any prime indices of the parts.

Views

Author