A316697 -id:A316697 - cp's OEIS frontend

A331683 One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.

1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 152, 172, 212, 214, 224, 256, 262, 304, 326, 344, 424, 428, 448, 512, 524, 526, 608, 622, 652, 688, 766, 848, 856, 886, 896, 1024, 1048, 1052, 1154, 1216, 1226, 1244, 1304, 1376, 1438, 1532, 1696

Offset: 1

Gus Wiseman, Jan 30 2020

nonn

Also Matula-Goebel numbers of lone-child-avoiding rooted trees at with at most one non-leaf branch under any given vertex. A rooted tree is lone-child-avoiding if 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 the root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Also Matula-Goebel numbers of lone-child-avoiding locally disjoint semi-identity trees. 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 sequence of all 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
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  212: (oo(oooo))
  214: (o(oo(oo)))
  224: (ooooo(oo))

Robert Israel, Table of n, a(n) for n = 1..3140

These trees counted by number of vertices are A212804.
The semi-lone-child-avoiding version is A331681.
The non-semi-identity version is A331871.
Lone-child-avoiding rooted trees are counted by A001678.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Unlabeled semi-identity trees are counted by A306200, with Matula-Goebel numbers A306202.
Locally disjoint rooted trees are counted by A316473.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Lone-child-avoiding locally disjoint rooted trees by leaves are A316697.
Cf. A000081, A007097, A061775, A196050, A276625, A316470, A316696, A331678, A331679, A331680, A331682, A331686, A331687.

N:= 10^4: # for terms <= N
S:= {1}:
with(queue):
Q:= new(1):
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,k_}}/;k>1)];
Select[Range[100],uryQ]

Intersection of A291636, A316495, and A306202.

A050381 Number of series-reduced planted trees with n leaves of 2 colors.

Original entry on oeis.org

2, 3, 10, 40, 170, 785, 3770, 18805, 96180, 502381, 2667034, 14351775, 78096654, 429025553, 2376075922, 13252492311, 74372374366, 419651663108, 2379399524742, 13549601275893, 77460249369658, 444389519874841

Offset: 1

Christian G. Bower, Nov 15 1999

nonn

Consider the free algebraic system with two commutative associative operators (x+y) and (x*y) and two generators A,B. The number of elements with n occurrences of the generators is 2*a(n) if n>1, and the number of generators if n=1. - Michael Somos, Aug 07 2017
From Gus Wiseman, Feb 07 2020: (Start)
Also the number of semi-lone-child-avoiding rooted trees with n leaves. Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf. For example, the a(1) = 2 through a(3) = 10 trees are:
  o    (oo)      (ooo)
  (o)  (o(o))    (o(oo))
       ((o)(o))  (oo(o))
                 ((o)(oo))
                 (o(o)(o))
                 (o(o(o)))
                 ((o)(o)(o))
                 ((o)(o(o)))
                 (o((o)(o)))
                 ((o)((o)(o)))
(End)

			For n=2, the 2*a(2) = 6 elements are: A+A, A+B, B+B, A*A, A*B, B*B. - _Michael Somos_, Aug 07 2017

Andrew Howroyd, Table of n, a(n) for n = 1..500
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
F. Chapoton, F. Hivert, J.-C. Novelli, A set-operad of formal fractions and dendriform-like sub-operads, arXiv preprint arXiv:1307.0092 [math.CO], 2013.
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012. - From _N. J. A. Sloane_, Dec 22 2012
N. J. A. Sloane, Transforms
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Index entries for sequences related to rooted trees

Column 2 of A319254.
Cf. A029856, A031148.
Lone-child-avoiding rooted trees with n leaves are A000669.
Lone-child-avoiding rooted trees with n vertices are A001678.
The locally disjoint case is A331874.
Semi-lone-child-avoiding rooted trees with n vertices are A331934.
Matula-Goebel numbers of these trees are A331935.
Cf. A000081, A005804, A141268, A196545, A291636, A316697, A330465, A331872, A331933, A331964.

terms = 22;
B[x_] = x O[x]^(terms+1);
A[x_] = 1/(1 - x + B[x])^2;
Do[A[x_] = A[x]/(1 - x^k + B[x])^Coefficient[A[x], x, k] + O[x]^(terms+1) // Normal, {k, 2, terms+1}];
Join[{2}, Drop[CoefficientList[A[x], x]/2, 2]] (* Jean-François Alcover, Aug 17 2018, after Michael Somos *)
slaurte[n_]:=If[n==1,{o,{o}},Join@@Table[Union[Sort/@Tuples[slaurte/@ptn]],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[slaurte[n]],{n,10}] (* Gus Wiseman, Feb 07 2020 *)

{a(n) = my(A, B); if( n<2, 2*(n>0), B = x * O(x^n); A = 1 / (1 - x + B)^2; for(k=2, n, A /= (1 - x^k + B)^polcoeff(A, k)); polcoeff(A, n)/2)}; /* Michael Somos, Aug 07 2017 */

Doubles (index 2+) under EULER transform.
Product_{k>=1} (1-x^k)^-a(k) = 1 + a(1)*x + Sum_{k>=2} 2*a(k)*x^k. - Michael Somos, Aug 07 2017
a(n) ~ c * d^n / n^(3/2), where d = 6.158893517087396289837838459951206775682824030495453326610366016992093939... and c = 0.1914250508201011360729769525164141605187995730026600722369002... - Vaclav Kotesovec, Aug 17 2018

A316694 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 28, 62, 143, 338, 804, 1948, 4789, 11886, 29796, 75316, 191702, 491040, 1264926, 3274594, 8514784, 22229481, 58243870

Offset: 1

Gus Wiseman, Jul 10 2018

A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root. It is an identity tree if no branch appears multiple times under the same root.

			The a(7) = 28 rooted trees:
  7,
  (16),
  (25),
  (1(15)),
  (34),
  (1(24)), (2(14)), (4(12)), (124),
  (1(1(14))),
  (3(13)),
  (2(23)),
  (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), (12(13)), (13(12)),
  (1(1(1(13)))),
  (2(2(12))),
  (1(1(2(12)))), (1(2(1(12)))), (1(12(12))), (2(1(1(12)))), (12(1(12))),
  (1(1(1(1(12))))).
Missing from this list but counted by A300660 are ((12)(13)) and ((12)(1(12))).

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A000081, A000669, A001678, A141268, A292504, A316653, A316654, A316656.
The semi-identity tree version is A212804.
Not requiring local disjointness gives A300660.
The non-identity tree version is A316696.
This is the case of A331686 where all leaves are singletons.
Rooted identity trees are A004111.
Locally disjoint rooted identity trees are A316471.
Lone-child-avoiding locally disjoint rooted trees are A331680.
Locally disjoint enriched identity p-trees are A331684.
Cf. A306200, A316697, A331678, A331679, A331681, A331683, A331783, A331874.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,disjointQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

a(21)-a(23) from Robert Price, Sep 16 2018

Updated with corrected terminology by Gus Wiseman, Feb 06 2020

A331686 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

Original entry on oeis.org

1, 2, 4, 8, 17, 41, 103, 280, 793, 2330, 6979, 21291

Offset: 1

Gus Wiseman, Jan 31 2020

A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (unequal) child of the same vertex. Lone-child-avoiding means there are no unary branchings. In an identity tree, all branches of any given vertex are distinct.

			The a(1) = 1 through a(5) = 17 trees:
  (1)  (2)   (3)       (4)            (5)
       (11)  (12)      (13)           (14)
             (111)     (22)           (23)
             ((1)(2))  (112)          (113)
                       (1111)         (122)
                       ((1)(3))       (1112)
                       ((2)(11))      (11111)
                       ((1)((1)(2)))  ((1)(4))
                                      ((2)(3))
                                      ((1)(22))
                                      ((3)(11))
                                      ((2)(111))
                                      ((1)((1)(3)))
                                      ((2)((1)(2)))
                                      ((11)((1)(2)))
                                      ((1)((2)(11)))
                                      ((1)((1)((1)(2))))

The non-identity version is A331678.
The case where the leaves are all singletons is A316694.
Identity trees are A004111.
Locally disjoint identity trees are A316471.
Locally disjoint enriched identity p-trees are A331684.
Lone-child-avoiding locally disjoint rooted semi-identity trees are A212804.
Cf. A000669, A001678, A005804, A141268, A300660, A316697, A319312, A331679, A331683, A331783, A331874, A331875.

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]]]];
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]],UnsameQ@@#&&disjointQ[#]&],{p,Select[mps[m],Length[#]>1&]}],m];
Table[Sum[Length[mpti[m]],{m,Sort/@IntegerPartitions[n]}],{n,8}]

A331873 Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 69, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 138, 144, 148, 152, 161, 162, 169, 172, 178, 184, 192, 196, 202, 206, 207, 208, 212, 214, 216, 224, 243

Offset: 1

Gus Wiseman, Feb 02 2020

nonn

First differs from A331936 in having 69, the Matula-Goebel number of the tree ((o)((o)(o))).
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 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, two, 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 semi-lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  46: (o((o)(o)))
  48: (oooo(o))
  49: ((oo)(oo))

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 A316495.
A superset of A320269.
The semi-identity tree case is A331681.
The non-semi version (i.e., not containing 2) is A331871.
These trees counted by vertices are A331872.
These trees counted by leaves are A331874.
Not requiring local disjointness gives A331935.
The identity tree case is A331937.
Cf. A007097, A050381, A061775, A196050, A291636, A302696, A316473, A316696, A316697, A331680, A331682, A331683, A331687, A331934.

msQ[n_]:=n==1||n==2||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],msQ]

A316696 Number of lone-child-avoiding locally disjoint rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 2, 4, 11, 27, 80, 218, 654, 1923, 5924, 18310, 58176, 186341, 606814, 1993420, 6618160, 22134640

Offset: 1

Gus Wiseman, Jul 10 2018

A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.

			The a(4) = 11 rooted trees:
  4,
  (13),
  (22),
  (1(12)), (2(11)), (112),
  (1(1(11))), (1(111)), ((11)(11)), (11(11)), (1111).

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Matula-Goebel numbers of locally disjoint rooted trees are A316495.
The case where all leaves are 1's is A316697.
Lone-child-avoiding locally disjoint rooted trees are A331680.
Cf. A000669, A001678, A141268, A316473, A316652, A331678, A331686, A331687, A331871, A331872, A331874.

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

a(16)-a(17) from Robert Price, Sep 16 2018

Terminology corrected by Gus Wiseman, Feb 06 2020

A331679 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.

Original entry on oeis.org

1, 2, 3, 8, 16, 48, 116, 341, 928, 2753, 7996, 24254, 73325, 226471, 702122

Offset: 1

Gus Wiseman, Jan 25 2020

A tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. It is lone-child-avoiding if there are no unary branchings.

			The a(1) = 1 through a(5) = 16 trees:
  1  2     3        4           5
     (11)  (111)    (22)        (11111)
           (1(11))  (1111)      ((11)3)
                    (2(11))     (1(22))
                    (1(111))    (2(111))
                    (11(11))    (1(1111))
                    ((11)(11))  (11(111))
                    (1(1(11)))  (111(11))
                                (1(2(11)))
                                (2(1(11)))
                                (1(1(111)))
                                (1(11)(11))
                                (1(11(11)))
                                (11(1(11)))
                                (1((11)(11)))
                                (1(1(1(11))))

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.

The non-locally disjoint version is A141268.
Locally disjoint trees counted by vertices are A316473.
The case where all leaves are 1's is A316697.
Number of trees counted by A331678 with all atoms equal to 1.
Matula-Goebel numbers of locally disjoint rooted trees are A316495.
Unlabeled lone-child-avoiding locally disjoint rooted trees are A331680.
Cf. A000081, A000669, A001678, A005804, A060356, A300660, A316471, A316694, A316696, A319312, A330465, A331681.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
usot[n_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[usot/@ptn]],disjointQ[DeleteCases[#,_?AtomQ]]&&SameQ@@Select[#,AtomQ]&],{ptn,Select[IntegerPartitions[n],Length[#]>1&]}],n];
Table[Length[usot[n]],{n,12}]

A331680 Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 45, 72, 124, 201, 341, 561, 947, 1571, 2651, 4434, 7496, 12631, 21423, 36332, 61910, 105641, 180924, 310548, 534713, 923047

Offset: 1

Gus Wiseman, Jan 25 2020

First differs from A320268 at a(11) = 45, A320268(11) = 44.

A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. Lone-child-avoiding means there are no unary branchings.

			The a(1) = 1 through a(9) = 16 trees (empty column indicated by dot):
  o  .  (oo)  (ooo)  (oooo)   (ooooo)   (oooooo)    (ooooooo)    (oooooooo)
                     (o(oo))  (o(ooo))  (o(oooo))   (o(ooooo))   (o(oooooo))
                              (oo(oo))  (oo(ooo))   (oo(oooo))   (oo(ooooo))
                                        (ooo(oo))   (ooo(ooo))   (ooo(oooo))
                                        ((oo)(oo))  (oooo(oo))   (oooo(ooo))
                                        (o(o(oo)))  (o(o(ooo)))  (ooooo(oo))
                                                    (o(oo)(oo))  ((ooo)(ooo))
                                                    (o(oo(oo)))  (o(o(oooo)))
                                                    (oo(o(oo)))  (o(oo(ooo)))
                                                                 (o(ooo(oo)))
                                                                 (oo(o(ooo)))
                                                                 (oo(oo)(oo))
                                                                 (oo(oo(oo)))
                                                                 (ooo(o(oo)))
                                                                 (o((oo)(oo)))
                                                                 (o(o(o(oo))))

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.

The enriched version is A316696.
The Matula-Goebel numbers of these trees are A331871.
The non-locally disjoint version is A001678.
These trees counted by number of leaves are A316697.
The semi-lone-child-avoiding version is A331872.
Cf. A000081, A000669, A005804, A060356, A141268, A300660, A316471, A316473, A316694, A316495, A319312, A330465, A331679, A331681, A331683.

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

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

Original entry on oeis.org

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}]

cp's OEIS Frontend

A331683 One and all numbers of the form 2^k * prime(j) for k > 0 and j already in the sequence.

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A050381 Number of series-reduced planted trees with n leaves of 2 colors.

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A316694 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.

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A331686 Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.

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A331873 Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.

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A316696 Number of lone-child-avoiding locally disjoint rooted trees whose leaves form an integer partition of n.

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A331679 Number of lone-child-avoiding locally disjoint rooted trees whose leaves are positive integers summing to n, with no two distinct leaves directly under the same vertex.

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A331680 Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.

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