A316494 -id:A316494 - cp's OEIS frontend

A316495 Matula-Goebel numbers of locally disjoint unlabeled rooted trees, meaning no branch overlaps any other (unequal) branch of the same root.

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 47, 48, 50, 51, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 77, 79, 80, 82, 85

Offset: 1

Gus Wiseman, Jul 04 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A number is in the sequence iff either it is equal to 1, it is a prime number whose prime index already belongs to the sequence, or its distinct prime indices are pairwise coprime and already belong to the sequence.

			The sequence of all locally disjoint rooted trees preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  16: (oooo)
  17: (((oo)))
  18: (o(o)(o))
  19: ((ooo))
  20: (oo((o)))

Cf. A000081, A007097, A302696, A303362, A304713, A316468, A316470, A316473, A316475, A316494.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
go[n_]:=Or[n==1,And[Or[PrimeQ[n],CoprimeQ@@Union[primeMS[n]]],And@@go/@primeMS[n]]];
Select[Range[100],go]

A316471 Number of locally disjoint rooted identity trees with n nodes, meaning no branch overlaps any other branch of the same root.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 21, 43, 89, 185, 391, 840, 1822, 3975, 8727, 19280, 42841, 95661, 214490

Offset: 1

Gus Wiseman, Jul 04 2018

			The a(7) = 11 locally disjoint 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)))

Gus Wiseman, The a(8) = 21 locally disjoint rooted identity trees.

Cf. A000081, A004111, A276625, A277098, A302696, A303362, A304713, A316467, A316471, A316473, A316474, A316494.

strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@#&&Select[Tuples[#,2],UnsameQ@@#&&(Intersection@@#=!={})&]=={}&]];
Table[Length[strut[n]],{n,20}]

A316467 Matula-Goebel numbers of locally stable rooted identity trees, meaning no branch is a subset of any other branch of the same root.

Original entry on oeis.org

1, 2, 3, 5, 11, 15, 31, 33, 47, 55, 93, 127, 137, 141, 155, 165, 211, 257, 341, 381, 411, 465, 487, 633, 635, 709, 771, 773, 811, 907, 977, 1023, 1055, 1285, 1297, 1397, 1457, 1461, 1507, 1621, 1705, 1905, 2127, 2293, 2319, 2321, 2433, 2621, 2721, 2833, 2931

Offset: 1

Gus Wiseman, Jul 04 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A number belongs to this sequence iff it is squarefree, its distinct prime indices are pairwise indivisible, and its prime indices also belong to this sequence.

			165 = prime(2)*prime(3)*prime(5) belongs to the sequence because it is squarefree, the indices {2,3,5} are pairwise indivisible, and each of them already belongs to the sequence.
Sequence of locally stable rooted identity trees preceded by their Matula-Goebel numbers begins:
    1: o
    2: (o)
    3: ((o))
    5: (((o)))
   11: ((((o))))
   15: ((o)((o)))
   31: (((((o)))))
   33: ((o)(((o))))
   47: (((o)((o))))
   55: (((o))(((o))))
   93: ((o)((((o)))))
  127: ((((((o))))))
  137: (((o)(((o)))))
  141: ((o)((o)((o))))
  155: (((o))((((o)))))
  165: ((o)((o))(((o))))

Cf. A000081, A004111, A007097, A276625, A277098, A285572, A285573, A302796, A303362, A304713, A316468, A316469, A316471, A316474, A316476, A316494.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
ain[n_]:=And[Select[Tuples[primeMS[n],2],UnsameQ@@#&&Divisible@@#&]=={},SquareFreeQ[n],And@@ain/@primeMS[n]];
Select[Range[100],ain]

A331783 Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.

Original entry on oeis.org

1, 1, 2, 4, 8, 17, 37, 83, 191, 450, 1076, 2610, 6404, 15875, 39676, 99880, 253016, 644524, 1649918, 4242226

Offset: 1

Gus Wiseman, Jan 31 2020

Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct.

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

The lone-child-avoiding case is A212804.
The identity tree version is A316471.
The Matula-Goebel numbers of these trees are given by A331682.
Identity trees are A004111.
Semi-identity trees are A306200.
Locally disjoint rooted trees are A316473.
Matula-Goebel numbers of locally disjoint semi-identity trees are A316494.
Cf. A000081, A306202, A316475, A316495, A316694, A331683, A331684, A331686.

disjunsQ[u_]:=Length[u]==1||UnsameQ@@DeleteCases[u,{}]&&Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldrsi[n_]:=If[n==1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[ldrsi/@c]]]/@IntegerPartitions[n-1],disjunsQ]];
Table[Length[ldrsi[n]],{n,10}]

A331682 One and all numbers whose prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be coprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 40, 41, 43, 44, 47, 48, 51, 52, 53, 55, 56, 58, 59, 60, 62, 64, 66, 67, 68, 70, 71, 74, 76, 77, 79, 80, 82, 85, 86, 88, 89, 93, 94, 95, 96, 101

Offset: 1

Gus Wiseman, Jan 27 2020

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.

Also Matula-Goebel numbers of locally disjoint rooted semi-identity trees. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches (of the root), which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of all locally disjoint rooted semi-identity trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  22: (o(((o))))
  24: (ooo(o))

The non-semi identity tree case is A316494.
The enumeration of these trees by vertices is A331783.
Semi-identity trees are counted by A306200.
Matula-Goebel numbers of semi-identity trees are A306202.
Locally disjoint rooted trees are counted by A316473.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Cf. A000081, A007097, A061775, A196050, A276625, A316470, A331681, A331683.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
deQ[n_]:=n==1||PrimeQ[n]&&deQ[PrimePi[n]]||CoprimeQ@@primeMS[n]&&And@@deQ/@primeMS[n];
Select[Range[100],deQ]

A331937 a(1) = 1; a(2) = 2; a(n + 1) = 2 * prime(a(n)).

Original entry on oeis.org

1, 2, 6, 26, 202, 2462, 43954, 1063462, 33076174, 1270908802, 58596709306, 3170266564862, 197764800466826, 14024066291995502, 1117378164606478094

Offset: 1

Gus Wiseman, Feb 07 2020

Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted identity trees. A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex. It is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf. In an identity tree, the branches of any given vertex are all distinct. The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of terms together with their associated trees begins:
     1: o
     2: (o)
     6: (o(o))
    26: (o(o(o)))
   202: (o(o(o(o))))
  2462: (o(o(o(o(o)))))

The semi-identity tree version is A331681.
Not requiring an identity tree gives A331873.
Not requiring local disjointness gives A331963.
Not requiring lone-child-avoidance gives A316494.
MG-numbers of semi-lone-child-avoiding rooted trees are A331935.
Cf. A007097, A061775, A276625, A316471, A316495, A316694, A331679, A331683, A331686, A331872, A331934, A331936, A331964, A331965.

msiQ[n_]:=n==1||n==2||!PrimeQ[n]&&SquareFreeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[1000],msiQ]

Intersection of A276625 (identity), A316495 (locally disjoint), and A331935 (semi-lone-child-avoiding).

a(14)-a(15) from Giovanni Resta, Feb 10 2020

A316469 Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

Original entry on oeis.org

1, 2, 6, 26, 78, 202, 606, 794, 2382, 2462, 2626, 7386, 7878, 8914, 10322, 12178, 26742, 30966, 32006, 36534, 42374, 43954, 47206, 80194, 96018, 115882, 127122, 131862, 141618, 149782, 158314, 160978, 184622, 217058, 240582, 248662, 260422, 347646, 449346

Offset: 1

Gus Wiseman, Jul 04 2018

nonn

A prime index of n is a number m such that prime(m) divides n. A number is in the sequence iff it is 1 or its prime indices are distinct, relatively prime, and already belong to the sequence.

			78 = prime(1)*prime(2)*prime(6) belongs to the sequence because the indices {1,2,6} are relatively prime, distinct, and already belong to the sequence.
The sequence of all identity RPMG-trees preceded by their Matula-Goebel numbers begins:
     1: o
     2: (o)
     6: (o(o))
    26: (o(o(o)))
    78: (o(o)(o(o)))
   202: (o(o(o(o))))
   606: (o(o)(o(o(o))))
   794: (o(o(o)(o(o))))
  2382: (o(o)(o(o)(o(o))))
  2462: (o(o(o(o(o)))))
  2626: (o(o(o))(o(o(o))))
  7386: (o(o)(o(o(o(o)))))
  7878: (o(o)(o(o))(o(o(o))))

Cf. A000081, A000837, A004111, A007097, A078374, A276625, A289509, A302696, A302796, A316467, A316470, A316471, A316474, A316494.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],Or[#==1,And[SquareFreeQ[#],GCD@@primeMS[#]==1,And@@#0/@primeMS[#]]]&]

A316503 Matula-Goebel numbers of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 41, 47, 55, 58, 62, 66, 78, 79, 82, 93, 94, 101, 109, 110, 113, 123, 127, 130, 137, 141, 143, 145, 155, 158, 165, 174, 179, 186, 195, 202, 205, 211, 218, 226, 246, 254, 257, 271, 274, 282, 286, 290, 293

Offset: 1

Gus Wiseman, Jul 05 2018

nonn

			Sequence of rooted identity trees preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   6: (o(o))
  10: (o((o)))
  11: ((((o))))
  13: ((o(o)))
  15: ((o)((o)))
  22: (o(((o))))
  26: (o(o(o)))
  29: ((o((o))))
  30: (o(o)((o)))
  31: (((((o)))))

Cf. A000081, A000837, A004111, A007097, A078374, A276625, A289509, A302796, A316467, A316470, A316494, A316502.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,And[SquareFreeQ[#],Or[PrimeQ[#],GCD@@primeMS[#]==1],And@@#0/@primeMS[#]]]&]

A319286 Number of series-reduced locally disjoint rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 2, 9, 67, 573, 6933, 97147, 1666999

Offset: 1

Gus Wiseman, Sep 16 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other branch of the same root.

			The a(3) = 9 trees:
  (1(11))
   (111)
  (1(12))
  (2(11))
   (112)
  (1(23))
  (2(13))
  (3(12))
   (123)
Examples of rooted trees that are not locally disjoint are ((11)(12)) and ((12)(13)).

Cf. A000081, A007562, A316473, A316475, A316494, A316495, A316496, A316497.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
gro[m_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],disjointQ]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]

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A316495 Matula-Goebel numbers of locally disjoint unlabeled rooted trees, meaning no branch overlaps any other (unequal) branch of the same root.

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A316471 Number of locally disjoint rooted identity trees with n nodes, meaning no branch overlaps any other branch of the same root.

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A316467 Matula-Goebel numbers of locally stable rooted identity trees, meaning no branch is a subset of any other branch of the same root.

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A331783 Number of locally disjoint rooted semi-identity trees with n unlabeled vertices.

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A331682 One and all numbers whose prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be coprime.

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A331937 a(1) = 1; a(2) = 2; a(n + 1) = 2 * prime(a(n)).

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A316469 Matula-Goebel numbers of unlabeled rooted identity RPMG-trees, meaning the Matula-Goebel numbers of the branches of any non-leaf node are relatively prime.

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A316503 Matula-Goebel numbers of unlabeled rooted identity trees with n nodes in which the branches of any node with more than one branch have empty intersection.

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A319286 Number of series-reduced locally disjoint rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

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