A279861 -id:A279861 - cp's OEIS frontend

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

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

A116549 a(0) = 1. a(m + 2^n) = a(n) + a(m), for 0 <= m <= 2^n - 1.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 7, 5, 6, 7, 8, 8, 9, 10, 11, 5, 6, 7, 8, 8, 9, 10, 11, 9, 10, 11, 12, 12, 13, 14, 15, 6, 7, 8, 9, 9, 10, 11, 12, 10, 11, 12, 13, 13, 14, 15, 16, 10, 11, 12, 13, 13, 14, 15, 16, 14, 15, 16, 17, 17, 18, 19, 20

Offset: 0

Leroy Quet, Mar 16 2006

Consider the following bijection between the natural numbers and hereditarily finite sets. For each n, write out n in binary. Assign to each set already given a natural number m the (m+1)-th digit of the binary number (reading from right to left). Let the set assigned to n contain all and only those sets which have a 1 for their digit. Then a(n) gives the number of pairs of braces appearing in the n-th set written out in full, e.g., for 3, we have {{{}}{}}, with 4 pairs of braces. - Thomas Anton, Mar 16 2019

			From _Gus Wiseman_, Jul 22 2019: (Start)
A finitary (or hereditarily finite) set is equivalent to a rooted identity tree. The following list shows the first few rooted identity trees together with their corresponding index in the sequence (o = leaf).
   0: o
   1: (o)
   2: ((o))
   3: (o(o))
   4: (((o)))
   5: (o((o)))
   6: ((o)((o)))
   7: (o(o)((o)))
   8: ((o(o)))
   9: (o(o(o)))
  10: ((o)(o(o)))
  11: (o(o)(o(o)))
  12: (((o))(o(o)))
  13: (o((o))(o(o)))
  14: ((o)((o))(o(o)))
  15: (o(o)((o))(o(o)))
  16: ((((o))))
  17: (o(((o))))
  18: ((o)(((o))))
  10: (o(o)(((o))))
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000523, A053645.

Cf. A000081, A000120, A004111, A029931, A048793, A061775, A070939, A072639, A276625, A279861, A326031.

import Data.Function (on); import Data.List (genericIndex)
a116549 = genericIndex a116549_list
a116549_list = 1 : zipWith ((+) `on` a116549) a000523_list a053645_list
-- Reinhard Zumkeller, Aug 27 2014

Nest[Append[#1, #1[[#3 + 1]] + #1[[#2 - 2^#3 + 1]] & @@ {#1, #2, Floor@ Log2@ #2}] & @@ {#, Length@ #} &, {1}, 63] (* Michael De Vlieger, Apr 21 2019 *)
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
dab[n_]:=1+Total[dab/@(bpe[n]-1)];
Array[dab,30,0] (* Gus Wiseman, Jul 22 2019 *)

For n > 0: a(n) = a(A000523(n)) + a(A053645(n)). - Reinhard Zumkeller, Aug 27 2014

A318187 Number of totally transitive rooted trees with n leaves.

Original entry on oeis.org

2, 2, 4, 8, 16, 32, 62, 122, 234, 451, 857, 1630, 3068, 5772, 10778, 20093, 37259

Offset: 1

Gus Wiseman, Aug 20 2018

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.

			The a(5) = 16 totally transitive rooted trees with 5 leaves:
  (o(o)(o(o)(o)))
  (o(o)(o)(o(o)))
  (o(o)(o)(o)(o))
  (o(o)(oo(o)))
  (oo(o)(o(o)))
  (o(o)(o)(oo))
  (oo(o)(o)(o))
  (o(o)(ooo))
  (o(oo)(oo))
  (oo(o)(oo))
  (ooo(o)(o))
  (o(oooo))
  (oo(ooo))
  (ooo(oo))
  (oooo(o))
  (ooooo)

Cf. A000081, A000669, A001678, A004111, A050381, A279861, A290689, A290760, A290822, A318185, A318186.

totralv[n_]:=totralv[n]=If[n==1,{{},{{}}},Join@@Table[Select[Union[Sort/@Tuples[totralv/@c]],Complement[Union@@#,#]=={}&],{c,Select[IntegerPartitions[n],Length[#]>1&]}]];
Table[Length[totralv[n]],{n,8}]

A279863 Number of maximal transitive finitary sets with n brackets.

Original entry on oeis.org

0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 2, 2, 1, 1, 4, 3, 4, 2, 5, 6, 10, 8, 11, 11, 20, 22, 29, 36, 45, 53, 77, 83, 108, 141, 172, 208, 274, 323

Offset: 1

Gus Wiseman, Dec 21 2016

nonn

A finitary set is transitive if every element is also a subset. A set system is maximal if the union is also a member.

			The a(23)=3 maximal transitive finitary sets are:
(()(())(()(()))((())(()(())))(()(())(()(())))),
(()(())((()))(((())))(()((())))(()(())((())))),
(()(())((()))(()(()))(()((())))(()(())((())))).

Gus Wiseman, Maximal transitive identity trees up to n=25

Cf. A001192, A004111, A276625, A279065, A279614, A279861.

maxtransfins[n_]:=If[n===1,{},Select[Union@@FixedPointList[Complement[Union@@Function[fin,Cases[Complement[Subsets[fin],fin],sub_:>With[{nov=Sort[Append[fin,sub]]},nov/;Count[Union[nov,{Union@@nov}],_List,{0,Infinity}]<=n]]]/@#,#]&,{{}}],And[Count[#,_List,{0,Infinity}]===n,MemberQ[#,Union@@#]]&]];
Table[Length[maxtransfins[n]],{n,20}]

A324842 Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 24, 28, 32, 36, 48, 54, 56, 64, 72, 78, 84, 96, 108, 112, 128, 144, 152, 156, 162, 168, 192, 196, 216, 224, 234, 252, 256, 288, 304, 312, 324, 336, 384, 392, 432, 444, 448, 456, 468, 486, 504, 512, 576, 588, 608, 624, 648, 672, 702

Offset: 1

Gus Wiseman, Mar 18 2019

nonn

			The sequence of rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  48: (oooo(o))
  54: (o(o)(o)(o))
  56: (ooo(oo))
  64: (oooooo)
  72: (ooo(o)(o))
  78: (o(o)(o(o)))
  84: (oo(o)(oo))
  96: (ooooo(o))

A subsequence of A120383.
Cf. A000081, A007097, A112798, A279861, A290689, A290760, A290822, A318186.
Cf. A324704, A324736, A324748, A324753, A324843, A324847, A324848, A324854.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
qaQ[n_]:=And[And@@Table[Divisible[n,x],{x,primeMS[n]}],And@@qaQ/@primeMS[n]];
Select[Range[1000],qaQ]

cp's OEIS Frontend

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

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A116549 a(0) = 1. a(m + 2^n) = a(n) + a(m), for 0 <= m <= 2^n - 1.

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A318187 Number of totally transitive rooted trees with n leaves.

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A279863 Number of maximal transitive finitary sets with n brackets.

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A324842 Matula-Goebel numbers of rooted trees where the branches of any branch of any terminal subtree form a submultiset of the branches of the same subtree.

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Mathematica