A324751 -id:A324751 - cp's OEIS frontend

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

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

Offset: 1

Gus Wiseman, Mar 13 2019

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

			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)))))

Gus Wiseman, The a(7) = 36 anti-transitive rooted trees.
Gus Wiseman, The a(10) = 532 anti-transitive rooted trees.

Cf. A276625, A279861, A279861, A290689, A290760, A304360.

Cf. A324694, A324695, A324738, A324741, A324743, A324751, A324754, A324756, A324758, A324759, A324764.

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

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

A324758 Heinz numbers of integer partitions containing no prime indices of the parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 37, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 57, 59, 61, 62, 63, 64, 65, 67, 68, 71, 73, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101

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).
These could be described as anti-transitive numbers (cf. A290822), as they are numbers x such that if prime(y) divides x and prime(z) divides y, then prime(z) does not divide x.
Also numbers n such that A003963(n) is coprime to n.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  27: {2,2,2}

The subset version is A324741, with maximal case A324743. The strict integer partition version is A324751. The integer partition version is A324756. An infinite version is A324695.

Cf. A000720, A001221, A001462, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324742, A324753, A324764.

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

A324756 Number of integer partitions of n containing no prime indices of the parts.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 7, 7, 9, 11, 16, 16, 24, 25, 34, 39, 50, 54, 70, 79, 96, 111, 135, 152, 186, 208, 249, 285, 335, 377, 448, 506, 588, 664, 777, 873, 1010, 1139, 1309, 1471, 1697, 1890, 2175, 2435, 2772, 3106, 3532, 3941, 4478, 4995, 5643, 6297, 7107, 7897

Offset: 0

Gus Wiseman, Mar 17 2019

nonn

These could be described as anti-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.

			The a(1) = 1 through a(8) = 9 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (43)       (44)
                    (31)    (11111)  (42)      (52)       (71)
                    (1111)           (51)      (331)      (422)
                                     (222)     (511)      (2222)
                                     (3111)    (31111)    (3311)
                                     (111111)  (1111111)  (5111)
                                                          (311111)
                                                          (11111111)

The subset version is A324741, with maximal case A324743. The strict case is A324751. The Heinz number version is A324758. An infinite version is A324695.

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

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

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

Gus Wiseman, Mar 17 2019

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}}}

			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))))))

Gus Wiseman, The a(9) = 41 anti-transitive rooted identity trees.

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.

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

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

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(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.

Andrew Howroyd, Table of n, a(n) for n = 0..100

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.

Table[Length[Select[Subsets[Range[n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,10}]

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(!bitand(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

A324765 Number of recursively anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 26, 52, 119, 266, 618, 1432, 3402, 8093, 19505, 47228, 115244, 282529, 696388, 1723400

Offset: 1

Gus Wiseman, Mar 17 2019

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 a(1) = 1 through a(6) = 11 recursively 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))))  (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Cf. A000081, A290689, A301700, A304360, A306844, A317787, A318185.
Cf. A324695, A324751, A324756, A324758, A324764, A324766, A324767, A324768.
Cf. A324838, A324840, A324844.

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

A324743 Number of maximal subsets of {1...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 8, 8, 8, 8, 12, 12, 18, 18, 19, 19, 30, 30, 54, 54, 54, 54, 96, 96, 96, 96, 96, 96, 156, 156, 244, 244, 248, 248, 248, 248, 440, 440, 440, 440, 688, 688, 1120, 1120, 1120, 1120, 1864, 1864, 1864, 1864, 1864, 1864, 3664, 3664, 3664, 3664, 3664

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(0) = 1 through a(8) = 8 maximal subsets:
  {}  {1}  {1}  {2}    {1,3}  {1,3}    {1,3}    {1,3,7}  {1,3,7}
           {2}  {1,3}  {2,4}  {1,5}    {1,5}    {1,5,7}  {1,5,7}
                       {3,4}  {3,4}    {2,4,5}  {2,4,5}  {2,4,5,8}
                              {2,4,5}  {3,4,6}  {2,5,7}  {2,5,7,8}
                                       {4,5,6}  {3,4,6}  {3,4,6,8}
                                                {3,6,7}  {3,6,7,8}
                                                {4,5,6}  {4,5,6,8}
                                                {5,6,7}  {5,6,7,8}
An example for n = 15 is {1,5,7,9,13,15}, with prime indices:
  1: {}
  5: {3}
  7: {4}
  9: {2,2}
  13: {6}
  15: {2,3}
None of these prime indices {2,3,4,6} belong to the subset, as required.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The non-maximal case is A324741. The case for subsets of {2...n} is A324763.
Cf. A000720, A001462, A007097, A084422, A085945, A112798, A276625, A290689, A290822, A304360, A306844, A320426, A324764.
Cf. A324695, A324736, A324742, A324744, A324751, A324756, A324758.

maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]],{n,0,10}]

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]));
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

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

Gus Wiseman, Mar 17 2019

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.

			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))))))

Gus Wiseman, The a(8) = 23 fully recursively anti-transitive rooted trees.
Gus Wiseman, The a(9) = 46 fully recursively anti-transitive rooted trees.
Gus Wiseman, The a(10) = 85 fully recursively anti-transitive rooted trees.

Cf. A000081, A279861, A290689, A304360, A306844, A318185.
Cf. A324695, A324751, A324756, A324758, A324763, A324765, A324768, A324769, A324770.
Cf. A324838, A324841, A324844, A324846.

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

A324768 Number of fully anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 11, 27, 60, 152, 376, 968, 2492, 6549, 17259, 46000, 123214, 332304, 900406, 2451999, 6703925

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.

			The a(1) = 1 through a(6) = 11 rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          ((o(o)))   ((o(oo)))
                          ((((o))))  ((oo(o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     (((o(o))))
                                     ((o((o))))
                                     (((((o)))))

Cf. A000081, A279861, A290689, A304360, A306844, A318185.
Cf. A324695, A324751, A324756, A324758, A324763, A324765, A324769, A324770.
Cf. A324838, A324840, A324844, A324846.

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

a(17)-a(20) from Jinyuan Wang, Jun 20 2020

A324742 Number of subsets of {2...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 2, 3, 6, 10, 16, 24, 48, 84, 144, 228, 420, 648, 1080, 1800, 3600, 5760, 11136, 16704, 31104, 53568, 90624, 136896, 269952, 515712, 862080, 1708800, 3171840, 4832640, 9325440, 14890752, 29781504, 52245504, 88418304, 166017024, 331628544, 497645568, 829409280

Offset: 1

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(1) = 1 through a(6) = 16 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}    {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 {4,5,6,12,17,18,19}, with prime indices:
   4: {1,1}
   5: {3}
   6: {1,2}
  12: {1,1,2}
  17: {7}
  18: {1,2,2}
  19: {8}
None of these prime indices {1,2,3,7,8} belong to the set, as required.

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

The maximal case is A324763. The version for subsets of {1...n} is A324741. The strict integer partition version is A324752. The integer partition version is A324757. The Heinz number version is A324761. An infinite version is A304360.
Cf. A000720, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A290689, A290822, A306844, A324764.
Cf. A324695, A324737, A324743, A324751, A324756, A324758.

Table[Length[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]));
((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

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

cp's OEIS Frontend

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

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A324758 Heinz numbers of integer partitions containing no prime indices of the parts.

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A324756 Number of integer partitions of n containing no prime indices of the parts.

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

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

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A324765 Number of recursively anti-transitive rooted trees with n nodes.

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A324743 Number of maximal subsets of {1...n} containing no prime indices of the elements.

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A324840 Number of fully recursively anti-transitive rooted trees with n nodes.

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