A316473 -id:A316473 - cp's OEIS frontend

A331681 One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.

1, 2, 4, 6, 8, 12, 14, 16, 24, 26, 28, 32, 38, 48, 52, 56, 64, 74, 76, 86, 96, 104, 106, 112, 128, 148, 152, 172, 178, 192, 202, 208, 212, 214, 224, 256, 262, 296, 304, 326, 344, 356, 384, 404, 416, 424, 428, 446, 448, 478, 512, 524, 526, 592, 608, 622, 652

Offset: 1

Gus Wiseman, Jan 26 2020

nonn

Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted semi-identity trees. A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. 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. Note that these conditions together imply that there is at most one non-leaf branch under any given vertex.
Also Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex.
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 semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex, together with their Matula-Goebel numbers, begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  38: (o(ooo))
  48: (oooo(o))
  52: (oo(o(o)))
  56: (ooo(oo))
  64: (oooooo)
  74: (o(oo(o)))
  76: (oo(ooo))
  86: (o(o(oo)))

Robert Israel, Table of n, a(n) for n = 1..4000

The enumeration of these trees by nodes is A324969 (essentially A000045).
The enumeration of these trees by leaves appears to be A090129(n + 1).
The (non-semi) lone-child-avoiding version is A331683.
Matula-Goebel numbers of rooted semi-identity trees are A306202.
Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
The set S of numbers with at most one prime index in S is A331784.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Cf. A000081, A000669, A001678, A007097, A061775, A196050, A291636, A316470, A316473, A331679, A331680, A331682, A331687, A331783, A331785, A331873.

N:= 1000: # for terms <= N
S:= {1,2}:
with(queue):
Q:= new(1,2):
while not empty(Q) do
  r:= dequeue(Q);
  p:= ithprime(r);
  newS:= {seq(2^i*p,i=1..ilog2(N/p))} minus S;
  S:= S union newS;
  for s in newS do enqueue(Q,s) od:
od:
sort(convert(S,list)); # Robert Israel, Feb 05 2020

uryQ[n_]:=n==1||MatchQ[FactorInteger[n],({{2,},{p,1}}/;uryQ[PrimePi[p]])|{{2,_}}];
Select[Range[100],uryQ]

Intersection of A306202 (semi-identity), A316495 (locally disjoint), and A331935 (semi-lone-child-avoiding). - Gus Wiseman, Jun 09 2020

A331872 Number of semi-lone-child-avoiding locally disjoint rooted trees with n vertices.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 12, 19, 35, 59, 104, 179, 318, 556, 993, 1772, 3202, 5807, 10643, 19594, 36380, 67915

Offset: 1

Gus Wiseman, Feb 02 2020

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.

Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.

			The a(1) = 1 through a(8) = 19 trees:
  o  (o)  (oo)  (ooo)   (oooo)    (ooooo)    (oooooo)     (ooooooo)
                (o(o))  (o(oo))   (o(ooo))   (o(oooo))    (o(ooooo))
                        (oo(o))   (oo(oo))   (oo(ooo))    (oo(oooo))
                        ((o)(o))  (ooo(o))   (ooo(oo))    (ooo(ooo))
                                  (o(o)(o))  (oooo(o))    (oooo(oo))
                                  (o(o(o)))  ((oo)(oo))   (ooooo(o))
                                             (o(o(oo)))   (o(o(ooo)))
                                             (o(oo(o)))   (o(oo)(oo))
                                             (oo(o)(o))   (o(oo(oo)))
                                             (oo(o(o)))   (o(ooo(o)))
                                             ((o)(o)(o))  (oo(o(oo)))
                                             (o((o)(o)))  (oo(oo(o)))
                                                          (ooo(o)(o))
                                                          (ooo(o(o)))
                                                          (o(o)(o)(o))
                                                          (o(o(o)(o)))
                                                          (o(o(o(o))))
                                                          (oo((o)(o)))
                                                          ((o)((o)(o)))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Not requiring lone-child-avoidance gives A316473.
The non-semi version is A331680.
The Matula-Goebel numbers of these trees are A331873.
The same trees counted by number of leaves are A331874.
Not requiring local disjointness gives A331934.
Lone-child-avoiding rooted trees are A001678.
Cf. A000081, A050381, A316696, A316697, A331678, A331679, A331681, A331686, A331687, A331871, A331935.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
strutsemi[n_]:=If[n==1,{{}},If[n==2,{{{}}},Select[Join@@Function[c,Union[Sort/@Tuples[strutsemi/@c]]]/@Rest[IntegerPartitions[n-1]],disjointQ]]];
Table[Length[strutsemi[n]],{n,8}]

A316468 Matula-Goebel numbers of locally stable rooted trees, meaning no branch is a submultiset of any other branch of the same root.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 15, 16, 17, 19, 23, 25, 27, 31, 32, 33, 35, 45, 47, 49, 51, 53, 55, 59, 64, 67, 69, 75, 77, 81, 83, 85, 93, 95, 97, 99, 103, 119, 121, 125, 127, 128, 131, 135, 137, 141, 149, 153, 155, 161, 165, 175, 177, 187, 197, 201, 207, 209

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 its distinct prime indices are pairwise indivisible and already belong to the sequence.

			Sequence of locally stable rooted trees preceded by their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  15: ((o)((o)))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))

Cf. A000081, A004111, A007097, A112798, A277098, A285572, A285573, A303362, A304713, A316467, A316470, A316473, A316475, A316476, A316495.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Or[#==1,And[Select[Tuples[primeMS[#],2],UnsameQ@@#&&Divisible@@#&]=={},And@@#0/@primeMS[#]]]&]

A331874 Number of semi-lone-child-avoiding locally disjoint rooted trees with n unlabeled leaves.

Original entry on oeis.org

2, 3, 8, 24, 67, 214, 687, 2406, 8672, 32641, 125431, 493039, 1964611

Offset: 1

Gus Wiseman, Feb 02 2020

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.

Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.

			The a(1) = 2 through a(4) = 24 trees:
  o    (oo)      (ooo)          (oooo)
  (o)  (o(o))    (o(oo))        (o(ooo))
       ((o)(o))  (oo(o))        (oo(oo))
                 (o(o)(o))      (ooo(o))
                 (o(o(o)))      ((oo)(oo))
                 ((o)(o)(o))    (o(o(oo)))
                 (o((o)(o)))    (o(oo(o)))
                 ((o)((o)(o)))  (oo(o)(o))
                                (oo(o(o)))
                                (o(o)(o)(o))
                                (o(o(o)(o)))
                                (o(o(o(o))))
                                (oo((o)(o)))
                                ((o)(o)(o)(o))
                                ((o(o))(o(o)))
                                ((oo)((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))))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Not requiring local disjointness gives A050381.
The non-semi version is A316697.
The same trees counted by number of vertices are A331872.
The Matula-Goebel numbers of these trees are A331873.
Lone-child-avoiding rooted trees counted  by leaves are A000669.
Semi-lone-child-avoiding rooted trees counted by vertices are A331934.
Cf. A000081, A001678, A300660, A316473, A316495, A316696, A331678, A331679, A331680, A331682, A331687, A331871, A331935.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
slaurt[n_]:=If[n==1,{o,{o}},Join@@Table[Select[Union[Sort/@Tuples[slaurt/@ptn]],disjointQ[Select[#,!AtomQ[#]&]]&],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[slaurt[n]],{n,8}]

A331687 Number of locally disjoint enriched p-trees of weight n.

Original entry on oeis.org

1, 2, 4, 12, 29, 93, 249, 803, 2337, 7480, 23130, 77372, 247598, 834507, 2762222

Offset: 1

Gus Wiseman, Jan 31 2020

A locally disjoint enriched p-tree of weight n is either the number n itself or a finite sequence of non-overlapping locally disjoint enriched p-trees whose weights are weakly decreasing and sum to n.

			The a(1) = 1 through a(4) = 12 enriched p-trees:
  1  2     3        4
     (11)  (21)     (22)
           (111)    (31)
           ((11)1)  (211)
                    (1111)
                    ((11)2)
                    ((21)1)
                    (2(11))
                    ((11)11)
                    ((111)1)
                    (((11)1)1)
                    ((11)(11))

The orderless version is A316696.
The identity case is A331684.
P-trees are A196545.
Enriched p-trees are A289501.
Locally disjoint identity trees are A316471.
Enriched identity p-trees are A331875.
Cf. A000669, A141268, A316473, A316495, A316694, A316697, A319312, A331678, A331679, A331680, A331686, A331871, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
ldep[n_]:=Prepend[Select[Join@@Table[Tuples[ldep/@p],{p,Rest[IntegerPartitions[n]]}],disjointQ[DeleteCases[#,_Integer]]&],n];
Table[Length[ldep[n]],{n,10}]

A331871 Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052

Offset: 1

Gus Wiseman, Feb 02 2020

nonn

First differs from A320269 in having 1589, the Matula-Goebel number of the tree ((oo)((oo)(oo))).
First differs from A331683 in having 49.
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.
Lone-child-avoiding means there are no unary branchings.
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.
Consists of one and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.

			The sequence of all lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))
The sequence of terms together with their prime indices begins:
     1: {}                  212: {1,1,16}
     4: {1,1}               214: {1,28}
     8: {1,1,1}             224: {1,1,1,1,1,4}
    14: {1,4}               256: {1,1,1,1,1,1,1,1}
    16: {1,1,1,1}           262: {1,32}
    28: {1,1,4}             304: {1,1,1,1,8}
    32: {1,1,1,1,1}         326: {1,38}
    38: {1,8}               343: {4,4,4}
    49: {4,4}               344: {1,1,1,14}
    56: {1,1,1,4}           361: {8,8}
    64: {1,1,1,1,1,1}       392: {1,1,1,4,4}
    76: {1,1,8}             424: {1,1,1,16}
    86: {1,14}              428: {1,1,28}
    98: {1,4,4}             448: {1,1,1,1,1,1,4}
   106: {1,16}              454: {1,49}
   112: {1,1,1,1,4}         512: {1,1,1,1,1,1,1,1,1}
   128: {1,1,1,1,1,1,1}     524: {1,1,32}
   152: {1,1,1,8}           526: {1,56}
   172: {1,1,14}            608: {1,1,1,1,1,8}
   196: {1,1,4,4}           622: {1,64}

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Not requiring local disjointness gives A291636.
Not requiring lone-child avoidance gives A316495.
A superset of A320269.
These trees are counted by A331680.
The semi-identity tree version is A331683.
The version containing 2 is A331873.
Cf. A001678, A007097, A050381, A061775, A196050, A302569, A302696, A316473, A316694, A316696, A316697, A331682, A331686, A331687, A331872, A331935.

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

Intersection of A291636 and A316495.

A143363 Number of ordered trees with n edges and having no protected vertices. A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.

Original entry on oeis.org

1, 1, 1, 3, 6, 17, 43, 123, 343, 1004, 2938, 8791, 26456, 80597, 247091, 763507, 2372334, 7413119, 23271657, 73376140, 232238350, 737638868, 2350318688, 7510620143, 24064672921, 77294975952, 248832007318, 802737926643

Offset: 0

Emeric Deutsch, Aug 20 2008

nonn

The "no protected vertices" condition can be rephrased as "every non-leaf vertex has at least one leaf child". But a(n) is also the number of ordered trees with n edges in which every non-leaf vertex has at most one leaf child. - David Callan, Aug 22 2014
Also the number of locally non-intersecting ordered rooted trees with n edges, meaning every non-leaf subtree has empty intersection. The unordered version is A007562. - Gus Wiseman, Nov 19 2022
a(n) is the number of parking functions of size n-1 avoiding the patterns 123, 132, and 213 . - Lara Pudwell, Apr 10 2023
For n>0, a(n) is the number of ways to place non-intersecting diagonals in convex n+3-gon so as to create no triangles such that none of the dividing diagonals passes through a chosen vertex. (empirical observation) - Muhammed Sefa Saydam, Feb 14 2025 and Aug 05 2025

			From _Gus Wiseman_, Nov 19 2022: (Start)
The a(0) = 1 through a(4) = 6 trees with at least one leaf directly under any non-leaf node:
  o  (o)  (oo)  (ooo)   (oooo)
                ((o)o)  ((o)oo)
                (o(o))  ((oo)o)
                        (o(o)o)
                        (o(oo))
                        (oo(o))
The a(0) = 1 through a(4) = 6 trees with at most one leaf directly under any node:
  o  (o)  ((o))  ((o)o)   (((o))o)
                 (o(o))   (((o)o))
                 (((o)))  ((o)(o))
                          ((o(o)))
                          (o((o)))
                          ((((o))))
(End)

Ayomikun Adeniran and Lara Pudwell, Pattern avoidance in parking functions, Enumer. Comb. Appl. 3:3 (2023), Article S2R17.
Gi-Sang Cheon and Louis W. Shapiro, Protected points in ordered trees, Appl. Math. Letters, 21, 2008, 516-520.
Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.

Cf. A143362.
For exactly one leaf directly under any node we have A006013.
The unordered version is A007562, ranked by A316470.
Allowing lone children gives A319378.
A000108 counts ordered rooted trees, unordered A000081.
A358453 counts transitive ordered trees, unordered A290689.
A358460 counts locally disjoint ordered trees, unordered A316473.
Cf. A318185, A324756, A324767, A324840, A324844.

p:=z^2*G^3-2*z*G^2-2*z^2*G^2+3*z*G+G+z^2*G-1-2*z=0: G:=RootOf(p,G): Gser:= series(G,z=0,33): seq(coeff(Gser,z,n),n=0..28);

a[n_Integer] := a[n] = Round[SeriesCoefficient[2 (x + 1 - Sqrt[x^2 - x + 1] Cos[ArcTan[(3 x Sqrt[12 x^3 - 96 x^2 - 24 x + 15])/(2 x^3 - 30 x^2 - 3 x + 2)]/3])/(3 x), {x, 0, n}]]; Table[a[n], {n, 0, 20}] (* Vladimir Reshetnikov, Apr 10 2022 *)
RecurrenceTable[{25 (n + 5) (n + 6) a[n + 5] - 10 (n + 5) (5 n + 21) a[n + 4] - 2 (77 n^2 + 613 n + 1185) a[n + 3] + 2 (50 n^2 + 253 n + 312) a[n + 2] + 4 (2 n + 1) (7 n + 9) a[n + 1] - 4 n (2 n + 1) a[n] == 0, a[0] == 1, a[1] == 1, a[2] == 1, a[3] == 3, a[4] == 6}, a[n], {n, 0, 27}] (* Vladimir Reshetnikov, Apr 11 2022 *)
ait[n_]:=ait[n]=If[n==1,{{}},Join@@Table[Select[Tuples[ait/@c],MemberQ[#,{}]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[ait[n]],{n,15}] (* Gus Wiseman, Nov 19 2022 *)

a(n) = A143362(n,0) for n>=1.
G.f.: G=G(z) satisfies z^2*G^3-2z(1+z)G^2+(1+3z+z^2)G-(1+2z)=0.
G.f.: (x+1-sqrt(x^2-x+1)*cos(arctan((3*x*sqrt(12*x^3-96*x^2-24*x+15))/(2*x^3-30*x^2-3*x+2))/3))*2/(3*x). - Vladimir Reshetnikov, Apr 10 2022
Recurrence: 25*(n+5)*(n+6)*a(n+5) - 10*(n+5)*(5*n+21)*a(n+4) - 2*(77*n^2+613*n+1185)*a(n+3) + 2*(50*n^2+253*n+312)*a(n+2) + 4*(2*n+1)*(7*n+9)*a(n+1) - 4*n*(2*n+1)*a(n) = 0. - Vladimir Reshetnikov, Apr 11 2022
From Muhammed Sefa Saydam, Jul 12 2025: (Start)
a(n) =  Sum_{k=2..n+2} A046736(k) * A046736(n-k+3) , for n >= 0 and A046736(1) = 1.
a(n) = A049125(n) + Sum_{k=1..n-2} A049125(k) * A046736(n-k+2), for n >= 3.
a(n) = A049125(n) + Sum_{k=1..n-2} a(k) * a(n-k-1), for n >= 3. (End)

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}]

A317787 Number of locally nonintersecting rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 8, 18, 40, 95, 227, 557, 1382, 3485, 8865, 22790, 59022, 153972, 404066, 1066236, 2826885, 7527411, 20121154

Offset: 1

Gus Wiseman, Aug 07 2018

An unlabeled rooted tree is locally nonintersecting if there is no common subbranch to all branches directly under any given node.

			The a(6) = 18 locally nonintersecting rooted trees:
  (((((o)))))
  ((((oo))))
  (((o(o))))
  ((o((o))))
  (o(((o))))
  ((o)((o)))
  (((ooo)))
  ((o(oo)))
  ((oo(o)))
  (o((oo)))
  (o(o(o)))
  (oo((o)))
  (o(o)(o))
  ((oooo))
  (o(ooo))
  (oo(oo))
  (ooo(o))
  (ooooo)
Missing from this list are (((o)(o))) and ((o)(oo)).

Cf. A000081, A276625, A301700, A316473, A316475, A316501, A316502, A317708, A317785, A317789.

rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,Intersection@@#=={}]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

a(16)-a(21) from Robert Price, Sep 16 2018

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]

cp's OEIS Frontend

A331681 One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.

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A331872 Number of semi-lone-child-avoiding locally disjoint rooted trees with n vertices.

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A316468 Matula-Goebel numbers of locally stable rooted trees, meaning no branch is a submultiset of any other branch of the same root.

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A331874 Number of semi-lone-child-avoiding locally disjoint rooted trees with n unlabeled leaves.

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A331687 Number of locally disjoint enriched p-trees of weight n.

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A331871 Matula-Goebel numbers of lone-child-avoiding locally disjoint rooted trees.

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A143363 Number of ordered trees with n edges and having no protected vertices. A protected vertex in an ordered tree is a vertex at least 2 edges away from its leaf descendants.

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

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