A291636 -id:A291636 - cp's OEIS frontend

A331489 Matula-Goebel numbers of topologically series-reduced rooted trees.

1, 2, 7, 8, 16, 19, 28, 32, 43, 53, 56, 64, 76, 98, 107, 112, 128, 131, 152, 163, 172, 196, 212, 224, 227, 256, 263, 266, 304, 311, 343, 344, 383, 392, 424, 428, 443, 448, 512, 521, 524, 532, 577, 602, 608, 613, 652, 686, 688, 719, 722, 742, 751, 784, 848, 856

Offset: 1

Gus Wiseman, Jan 20 2020

nonn

We say that a rooted tree is topologically series-reduced if no vertex (including the root) has degree 2.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.

			The sequence of all topologically series-reduced rooted trees together with their Matula-Goebel numbers begins:
    1: o
    2: (o)
    7: ((oo))
    8: (ooo)
   16: (oooo)
   19: ((ooo))
   28: (oo(oo))
   32: (ooooo)
   43: ((o(oo)))
   53: ((oooo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   98: (o(oo)(oo))
  107: ((oo(oo)))
  112: (oooo(oo))
  128: (ooooooo)
  131: ((ooooo))
  152: (ooo(ooo))
  163: ((o(ooo)))

Eric Weisstein's World of Mathematics, Series-reduced tree.
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Unlabeled rooted trees are counted by A000081.
Topologically series-reduced trees are counted by A000014.
Topologically series-reduced rooted trees are counted by A001679.
Labeled topologically series-reduced trees are counted by A005512.
Labeled topologically series-reduced rooted trees are counted by A060313.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000669, A001678, A007097, A007821, A060356, A061775, A109082, A109129, A196050, A254382, A276625, A330943, A331490.

nn=1000;
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
srQ[n_]:=Or[n==1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
Select[Range[nn],PrimeOmega[#]!=2&&And@@srQ/@primeMS[#]&]

A331578 Number of labeled series-reduced rooted trees with n vertices and more than two branches of the root.

Original entry on oeis.org

0, 0, 0, 4, 5, 186, 847, 17928, 166833, 3196630, 45667391, 925287276, 17407857337, 393376875906, 8989368580935, 229332484742416, 6094576250570849, 174924522900914094, 5271210321949744111, 168792243040279327860, 5674164658298121248361, 200870558472768096534490

Offset: 1

Gus Wiseman, Jan 21 2020

nonn

A rooted tree is series-reduced if no vertex (including the root) has degree 2.

Also labeled lone-child-avoiding rooted trees with n vertices and more than two branches, where a rooted tree is lone-child-avoiding if no vertex has exactly one child.

			Non-isomorphic representatives of the a(7) = 847 trees (in the format root[branches]) are:
  1[2,3,4[5,6,7]]
  1[2,3,4,5[6,7]]
  1[2,3,4,5,6,7]

Andrew Howroyd, Table of n, a(n) for n = 1..200

The non-series-reduced version is A331577.
The unlabeled version is A331488.
Lone-child-avoiding rooted trees are counted by A001678.
Topologically series-reduced rooted trees are counted by A001679.
Labeled topologically series-reduced rooted trees are counted by A060313.
Labeled lone-child-avoiding rooted trees are counted by A060356.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Matula-Goebel numbers of series-reduced rooted trees are A331489.
Cf. A000014, A000169, A000669, A005512, A108919, A206429, A331233, A331490.

lrt[set_]:=If[Length[set]==0,{},Join@@Table[Apply[root,#]&/@Join@@Table[Tuples[lrt/@stn],{stn,sps[DeleteCases[set,root]]}],{root,set}]];
Table[Length[Select[lrt[Range[n]],Length[#]>2&&FreeQ[#,[]]&]],{n,6}]

a(n) = {if(n<=1, 0, sum(k=1, n, (-1)^(n-k)*k^(k-2)*n*(n-2)!*binomial(n-1,k-1)*(2*k*n - n - k^2)/k!))} \\ Andrew Howroyd, Dec 09 2020

seq(n)={my(w=lambertw(-x/(1+x) + O(x*x^n))); Vec(serlaplace(-x - w - (x/2)*w^2), -n)} \\ Andrew Howroyd, Dec 09 2020

From Andrew Howroyd, Dec 09 2020: (Start)
a(n) = A060313(n) - n*A060356(n-1) for n > 1.
a(n) = Sum_{k=1..n} (-1)^(n-k)*k^(k-2)*n*(n-2)!*binomial(n-1,k-1)*(2*k*n - n - k^2)/k! for n > 1.
E.g.f.: -x - LambertW(-x/(1+x)) - (x/2)*LambertW(-x/(1+x))^2.
(End)

Terms a(9) and beyond from Andrew Howroyd, Dec 09 2020

A331967 Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 256, 343, 361, 512, 1024, 2048, 2401, 2809, 4096, 6859, 8192, 16384, 16807, 17161, 32768, 51529, 65536, 96721, 117649, 130321, 131072, 148877, 262144, 516961, 524288, 823543, 1048576, 2097152, 2248091, 2476099, 2621161, 4194304

Offset: 1

Gus Wiseman, Feb 06 2020

nonn

Lone-child-avoiding means there are no unary branchings.
In an achiral rooted tree, the branches of any given vertex are all equal.
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 numbers of the form prime(j)^k where k > 1 and j is already in the sequence.

			The sequence of all lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:
      1: o
      4: (oo)
      8: (ooo)
     16: (oooo)
     32: (ooooo)
     49: ((oo)(oo))
     64: (oooooo)
    128: (ooooooo)
    256: (oooooooo)
    343: ((oo)(oo)(oo))
    361: ((ooo)(ooo))
    512: (ooooooooo)
   1024: (oooooooooo)
   2048: (ooooooooooo)
   2401: ((oo)(oo)(oo)(oo))
   2809: ((oooo)(oooo))
   4096: (oooooooooooo)
   6859: ((ooo)(ooo)(ooo))
   8192: (ooooooooooooo)
  16384: (oooooooooooooo)
  16807: ((oo)(oo)(oo)(oo)(oo))
  17161: ((ooooo)(ooooo))
  32768: (ooooooooooooooo)
  51529: (((oo)(oo))((oo)(oo)))
  65536: (oooooooooooooooo)
  96721: ((oooooo)(oooooo))

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Gus Wiseman, The first 30 lone-child-avoiding achiral rooted trees.

A subset of A025475 (nonprime prime powers).
The enumeration of these trees by vertices is A167865.
Not requiring lone-child-avoidance gives A214577.
The semi-achiral version is A320269.
The semi-lone-child-avoiding version is A331992.
Achiral rooted trees are counted by A003238.
MG-numbers of planted achiral rooted trees are A280996.
MG-numbers of lone-child-avoiding rooted trees are A291636.
Cf. A001678, A007097, A061775, A196050, A276625, A291441, A320230, A320268, A331912, A331936, A331963, A331965, A331966, A331991.

msQ[n_]:=n==1||!PrimeQ[n]&&PrimePowerQ[n]&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[10000],msQ]

Intersection of A214577 (achiral) and A291636 (lone-child-avoiding).

A331994 Matula-Goebel numbers of semi-lone-child-avoiding rooted semi-identity trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 21, 24, 26, 28, 32, 38, 39, 42, 48, 52, 56, 57, 64, 74, 76, 78, 84, 86, 91, 96, 104, 106, 111, 112, 114, 128, 129, 133, 146, 148, 152, 156, 159, 168, 172, 178, 182, 192, 202, 208, 212, 214, 219, 222, 224, 228, 247, 256, 258, 259, 262

Offset: 1

Gus Wiseman, Feb 05 2020

nonn

Semi-lone-child-avoiding means there are no vertices with exactly one child unless that child is an endpoint/leaf.
In a semi-identity tree, the non-leaf branches of any given vertex are 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.
Consists of one, two, and all numbers that can be written as a power of two (other than 2) times a squarefree number whose prime indices already belong to the sequence, where a prime index of n is a number m such that prime(m) divides n.

			The sequence of all semi-lone-child-avoiding rooted semi-identity trees 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)
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  32: (ooooo)
  38: (o(ooo))
  39: ((o)(o(o)))
  42: (o(o)(oo))
  48: (oooo(o))
  52: (oo(o(o)))
  56: (ooo(oo))
  57: ((o)(ooo))
The sequence of terms together with their prime indices begins:
    1: {}              64: {1,1,1,1,1,1}      159: {2,16}
    2: {1}             74: {1,12}             168: {1,1,1,2,4}
    4: {1,1}           76: {1,1,8}            172: {1,1,14}
    6: {1,2}           78: {1,2,6}            178: {1,24}
    8: {1,1,1}         84: {1,1,2,4}          182: {1,4,6}
   12: {1,1,2}         86: {1,14}             192: {1,1,1,1,1,1,2}
   14: {1,4}           91: {4,6}              202: {1,26}
   16: {1,1,1,1}       96: {1,1,1,1,1,2}      208: {1,1,1,1,6}
   21: {2,4}          104: {1,1,1,6}          212: {1,1,16}
   24: {1,1,1,2}      106: {1,16}             214: {1,28}
   26: {1,6}          111: {2,12}             219: {2,21}
   28: {1,1,4}        112: {1,1,1,1,4}        222: {1,2,12}
   32: {1,1,1,1,1}    114: {1,2,8}            224: {1,1,1,1,1,4}
   38: {1,8}          128: {1,1,1,1,1,1,1}    228: {1,1,2,8}
   39: {2,6}          129: {2,14}             247: {6,8}
   42: {1,2,4}        133: {4,8}              256: {1,1,1,1,1,1,1,1}
   48: {1,1,1,1,2}    146: {1,21}             258: {1,2,14}
   52: {1,1,6}        148: {1,1,12}           259: {4,12}
   56: {1,1,1,4}      152: {1,1,1,8}          262: {1,32}
   57: {2,8}          156: {1,1,2,6}          266: {1,4,8}

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

The locally disjoint version is A331681.
The enumeration of these trees by vertices is A331993.
Semi-identity trees are A306200.
MG-numbers of rooted identity trees are A276625.
MG-numbers of lone-child-avoiding rooted identity trees are {1}.
MG-numbers of lone-child-avoiding rooted trees are A291636.
MG-numbers of semi-identity trees are A306202.
MG-numbers of semi-lone-child-avoiding rooted trees are A331935.
MG-numbers of semi-lone-child-avoiding rooted identity trees are A331963.
MG-numbers of lone-child-avoiding rooted semi-identity trees are A331965.
Cf. A004111, A007097, A061775, A122132, A196050, A320269, A331683, A331873, A331934, A331936, A331964, A331966.

scsiQ[n_]:=n==1||n==2||!PrimeQ[n]&&FreeQ[FactorInteger[n],{?(#>2&),?(#>1&)}]&&And@@scsiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],scsiQ]

Intersection of A306202 and A331935.

A298305 Matula-Goebel numbers of rooted trees with strictly thinning limbs.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 16, 18, 24, 27, 28, 32, 36, 42, 48, 52, 54, 56, 63, 64, 72, 78, 81, 84, 92, 96, 98, 104, 108, 112, 117, 126, 128, 138, 144, 147, 152, 156, 162, 168, 182, 184, 189, 192, 196, 207, 208, 216, 224, 228, 234, 243, 252, 256, 273, 276, 288, 294

Offset: 1

Gus Wiseman, Jan 16 2018

nonn

An unlabeled rooted tree has strictly thinning limbs if its outdegrees are strictly decreasing from root to leaves.

			Sequence of trees begins:
1  o
2  (o)
4  (oo)
6  (o(o))
8  (ooo)
9  ((o)(o))
12 (oo(o))
16 (oooo)
18 (o(o)(o))
24 (ooo(o))
27 ((o)(o)(o))
28 (oo(oo))
32 (ooooo)
36 (oo(o)(o))
42 (o(o)(oo))
48 (oooo(o))
52 (oo(o(o)))
54 (o(o)(o)(o))
56 (ooo(oo))
63 ((o)(o)(oo))
64 (oooooo)
72 (ooo(o)(o))
78 (o(o)(o(o)))
81 ((o)(o)(o)(o))
84 (oo(o)(oo))
92 (oo((o)(o)))
96 (ooooo(o))
98 (o(oo)(oo))

Cf. A000081, A007097, A061775, A111299, A124343, A124346, A214577, A276625, A290760, A291636, A298126, A298120, A298304.

MGtree[n_]:=If[n===1,{},MGtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
strthinQ[t_]:=And@@Cases[t,b_List:>Length[b]>Max@@Length/@b,{0,Infinity}];
Select[Range[200],strthinQ[MGtree[#]]&]

A318611 Number of series-reduced powerful rooted trees with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 1, 3, 3, 4, 4, 8, 5, 11, 10, 14, 14, 24, 18, 34, 32, 46, 45, 72, 60, 103, 96, 138, 137, 212, 184, 296, 282, 403, 397, 591, 539, 830, 798, 1125, 1119, 1624, 1519, 2253, 2195, 3067, 3056, 4341, 4158, 6004, 5897, 8145, 8164, 11397, 11090

Offset: 1

Gus Wiseman, Aug 30 2018

nonn

A series-reduced rooted tree is powerful if either it is a single node, or the branches of the root all appear with multiplicities greater than 1 and are themselves series-reduced powerful rooted trees.

			The a(13) = 8 series-reduced powerful rooted trees:
  ((oo)(oo)(oo)(oo))
  ((ooo)(ooo)(ooo))
  (ooo(oo)(oo)(oo))
  ((ooooo)(ooooo))
  (oo(oooo)(oooo))
  (oooo(ooo)(ooo))
  (oooooo(oo)(oo))
  (oooooooooooo)

Alois P. Heinz, Table of n, a(n) for n = 1..8000

Cf. A000081, A001190, A001678, A001694, A004111, A167865, A291636, A317102, A317705, A317707, A318612, A318691.

h:= proc(n, k, t) option remember; `if`(k=0, binomial(n+t, t),
      `if`(n=0, 0, add(h(n-1, k-j, t+1), j=2..k)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1)*h(a(i), j, 0), j=0..n/i)))
    end:
a:= n-> `if`(n<2, n, b(n-1$2)):
seq(a(n), n=1..60);  # Alois P. Heinz, Aug 31 2018

purt[n_]:=purt[n]=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],Min@@Length/@Split[#]>1&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[purt[n]],{n,20}]
(* Second program: *)
h[n_, k_, t_] := h[n, k, t] = If[k == 0, Binomial[n + t, t],
     If[n == 0, 0, Sum[h[n - 1, k - j, t + 1], {j, 2, k}]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     Sum[b[n - i*j, i - 1]*h[a[i], j, 0], {j, 0, n/i}]]];
a[n_] := If[n < 2, n, b[n - 1, n - 1]];
Array[a, 60] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(41)-a(56) from Alois P. Heinz, Aug 31 2018

A331992 Matula-Goebel numbers of semi-lone-child-avoiding achiral rooted trees.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 27, 32, 49, 64, 81, 128, 243, 256, 343, 361, 512, 529, 729, 1024, 2048, 2187, 2401, 2809, 4096, 6561, 6859, 8192, 10609, 12167, 16384, 16807, 17161, 19683, 32768, 51529, 59049, 65536, 96721, 117649, 130321, 131072, 148877, 175561, 177147

Offset: 1

Gus Wiseman, Feb 06 2020

nonn

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.
In an achiral rooted tree, the branches of any given vertex are all equal.
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 numbers of the form prime(j)^k where k > 1 and j is already in the sequence.

			The sequence of all semi-lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:
     1: o
     2: (o)
     4: (oo)
     8: (ooo)
     9: ((o)(o))
    16: (oooo)
    27: ((o)(o)(o))
    32: (ooooo)
    49: ((oo)(oo))
    64: (oooooo)
    81: ((o)(o)(o)(o))
   128: (ooooooo)
   243: ((o)(o)(o)(o)(o))
   256: (oooooooo)
   343: ((oo)(oo)(oo))
   361: ((ooo)(ooo))
   512: (ooooooooo)
   529: (((o)(o))((o)(o)))
   729: ((o)(o)(o)(o)(o)(o))
  1024: (oooooooooo)

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Gus Wiseman, The first 36 semi-lone-child-avoiding achiral rooted trees.

Except for two, a subset of A025475 (nonprime prime powers).
Not requiring achirality gives A331935.
The semi-achiral version is A331936.
The fully-chiral version is A331963.
The semi-chiral version is A331994.
The non-semi version is counted by A331967.
The enumeration of these trees by vertices is A331991.
Achiral rooted trees are counted by A003238.
MG-numbers of achiral rooted trees are A214577.
Cf. A001678, A007097, A050381, A061775, A167865, A196050, A276625, A280996, A291441, A291636, A320230, A320269, A331912, A331933, A331965.

msQ[n_]:=n<=2||!PrimeQ[n]&&Length[FactorInteger[n]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[10000],msQ]

Intersection of A214577 (achiral) and A331935 (semi-lone-child-avoiding).

A339193 Matula-Goebel numbers of unlabeled binary rooted semi-identity trees.

Original entry on oeis.org

1, 4, 14, 86, 301, 886, 3101, 3986, 13766, 13951, 19049, 48181, 57026, 75266, 85699, 199591, 263431, 295969, 298154, 302426, 426058, 882899

Offset: 1

Gus Wiseman, Mar 14 2021

nonn

Definition: A positive integer belongs to the sequence iff it is 1, 4, or a squarefree semiprime whose prime indices both already belong to the sequence. 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.
In a semi-identity tree, only the non-leaf branches of any given vertex are distinct. Alternatively, 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 an unlabeled 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 the corresponding unlabeled rooted trees begins:
      1: o
      4: (oo)
     14: (o(oo))
     86: (o(o(oo)))
    301: ((oo)(o(oo)))
    886: (o(o(o(oo))))
   3101: ((oo)(o(o(oo))))
   3986: (o((oo)(o(oo))))
  13766: (o(o(o(o(oo)))))
  13951: ((oo)((oo)(o(oo))))
  19049: ((o(oo))(o(o(oo))))
  48181: ((oo)(o(o(o(oo)))))
  57026: (o((oo)(o(o(oo)))))
  75266: (o(o((oo)(o(oo)))))
  85699: ((o(oo))((oo)(o(oo))))

Gus Wiseman, The sequence of all unlabeled binary rooted semi-identity trees by Matula-Goebel number.

Counting these trees by number of nodes gives A063895.
A000081 counts unlabeled rooted trees with n nodes.
A111299 ranks binary trees, counted by A001190.
A276625 ranks identity trees, counted by A004111.
A306202 ranks semi-identity trees, counted by A306200.
A306203 ranks balanced semi-identity trees, counted by A306201.
A331965 ranks lone-child avoiding semi-identity trees, counted by A331966.
Cf. A007097, A061775, A196050, A291636, A331963, A331964.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mgbiQ[n_]:=Or[n==1,n==4,SquareFreeQ[n]&&PrimeOmega[n]==2&&And@@mgbiQ/@primeMS[n]];
Select[Range[1000],mgbiQ]

A298126 Matula-Goebel numbers of rooted trees in which all outdegrees are even.

Original entry on oeis.org

1, 4, 14, 16, 49, 56, 64, 86, 106, 196, 224, 256, 301, 344, 371, 424, 454, 526, 622, 686, 784, 886, 896, 1024, 1154, 1204, 1376, 1484, 1589, 1696, 1816, 1841, 1849, 2104, 2177, 2279, 2386, 2401, 2488, 2744, 2809, 2846, 3101, 3136, 3238, 3544, 3584, 3986, 4039

Offset: 1

Gus Wiseman, Jan 13 2018

nonn

			Sequence of trees begins:
1   o
4   (oo)
14  (o(oo))
16  (oooo)
49  ((oo)(oo))
56  (ooo(oo))
64  (oooooo)
86  (o(o(oo)))
106 (o(oooo))
196 (oo(oo)(oo))
224 (ooooo(oo))
256 (oooooooo)
301 ((oo)(o(oo)))
344 (ooo(o(oo)))
371 ((oo)(oooo))
424 (ooo(oooo))
454 (o((oo)(oo)))
526 (o(ooo(oo)))
622 (o(oooooo))
686 (o(oo)(oo)(oo))
784 (oooo(oo)(oo))
886 (o(o(o(oo))))
896 (ooooooo(oo))

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A001190, A007097, A026424, A028260, A061775, A111299, A214577, A245824, A276625, A277098, A290760, A291441, A291442, A291636, A295461, A297571, A298118, A298120.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
etQ[n_]:=Or[n===1,With[{m=primeMS[n]},EvenQ@Length@m&&And@@etQ/@m]];
Select[Range[10000],etQ]

A298303 Matula-Goebel numbers of rooted trees with thinning limbs.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 35, 36, 39, 40, 42, 44, 45, 46, 48, 49, 50, 52, 54, 55, 56, 58, 60, 62, 63, 64, 65, 66, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 86, 87, 88, 90, 91, 92, 93, 94

Offset: 1

Gus Wiseman, Jan 16 2018

nonn

An unlabeled rooted tree has thinning limbs if its outdegrees are weakly decreasing from root to leaves.

Cf. A000081, A007097, A061775, A111299, A124343, A124346, A214577, A276625, A290760, A291636, A298126, A298120, A298304, A298305.

MGtree[n_]:=If[n===1,{},MGtree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
thinQ[t_]:=And@@Cases[t,b_List:>Length[b]>=Max@@Length/@b,{0,Infinity}];
Select[Range[200],thinQ[MGtree[#]]&]

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