A291442 -id:A291442 - cp's OEIS frontend

A291636 Matula-Goebel numbers of lone-child-avoiding rooted trees.

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 133, 152, 172, 196, 212, 214, 224, 256, 262, 266, 301, 304, 326, 343, 344, 361, 371, 392, 424, 428, 448, 454, 512, 524, 526, 532, 602, 608, 622, 652, 686, 688, 722, 742, 749, 766, 784, 817

Offset: 1

Gus Wiseman, Aug 28 2017

nonn

We say that a rooted tree is lone-child-avoiding if no vertex has exactly one child.
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.
An alternative definition: n is in the sequence iff n is 1 or the product of two or more not necessarily distinct prime numbers whose prime indices already belong to the sequence. For example, 14 is in the sequence because 14 = prime(1) * prime(4) and 1 and 4 both already belong to the sequence.

			The sequence of all lone-child-avoiding 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)
  133: ((oo)(ooo))
  152: (ooo(ooo))
  172: (oo(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.
Index entries for sequences related to Matula-Goebel numbers

These trees are counted by A001678.
The case with more than two branches is A331490.
Unlabeled rooted trees are counted by A000081.
Topologically series-reduced rooted trees are counted by A001679.
Labeled lone-child-avoiding rooted trees are counted by A060356.
Labeled lone-child-avoiding unrooted trees are counted by A108919.
MG numbers of singleton-reduced rooted trees are A330943.
MG numbers of topologically series-reduced rooted trees are A331489.
Cf. A007097, A061775, A109082, A109129, A111299, A196050, A198518, A276625, A291441, A291442, A331488.

nn=2000;
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],srQ]

Updated with corrected terminology by Gus Wiseman, Jan 20 2020

A303431 Aperiodic tree numbers. Matula-Goebel numbers of aperiodic rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 12, 13, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 37, 39, 40, 41, 44, 45, 47, 48, 50, 52, 54, 55, 58, 60, 61, 62, 65, 66, 71, 72, 74, 75, 78, 79, 80, 82, 87, 88, 89, 90, 93, 94, 96, 99, 101, 104, 108, 109, 110, 111, 113, 116, 117, 120, 122

Offset: 1

Gus Wiseman, Apr 23 2018

nonn

A positive integer is an aperiodic tree number iff either it is equal to 1 or it belongs to A007916 (numbers that are not perfect powers, or numbers whose prime multiplicities are relatively prime) and all of its prime indices are also aperiodic tree numbers, where a prime index of n is a number m such that prime(m) divides n.

			Sequence of aperiodic rooted trees begins:
01 o
02 (o)
03 ((o))
05 (((o)))
06 (o(o))
10 (o((o)))
11 ((((o))))
12 (oo(o))
13 ((o(o)))
15 ((o)((o)))
18 (o(o)(o))
20 (oo((o)))
22 (o(((o))))
24 (ooo(o))
26 (o(o(o)))
29 ((o((o))))
30 (o(o)((o)))
31 (((((o)))))
33 ((o)(((o))))

Cf. A000081, A000740, A000837, A007097, A007916, A052409, A052410, A061775, A111299, A214577, A275024, A276625, A290760, A290822, A291442, A298533, A298536, A301700, A302242, A303386.

zapQ[1]:=True;zapQ[n_]:=And[GCD@@FactorInteger[n][[All,2]]===1,And@@zapQ/@PrimePi/@FactorInteger[n][[All,1]]];
Select[Range[100],zapQ]

A291443 Number of leaf-balanced trees with n nodes.

Original entry on oeis.org

1, 1, 2, 4, 8, 14, 25, 41, 70, 116, 198, 331, 568, 957, 1635, 2776, 4757, 8144, 14089, 24428, 42707, 74895, 131983, 232895, 411725, 727434, 1284809, 2265997, 3992154, 7023718, 12347202, 21690274, 38096244, 66915426, 117591030, 206791336, 364037186, 641690280

Offset: 1

Gus Wiseman, Aug 23 2017

nonn

An unlabeled rooted tree is leaf-balanced if every branch has the same number of leaves and every non-leaf rooted subtree is also leaf-balanced.

			The a(5)=8 leaf-balanced trees are: ((((o)))), (((oo))), ((o(o))), ((ooo)), (o((o))), ((o)(o)), (oo(o)), (oooo). The tree (o(oo)) is not leaf-balanced.

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A000081, A003238, A291442.

allbal[n_]:=allbal[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[allbal/@c]],SameQ@@(Count[#,{},{0,Infinity}]&/@#)&]]/@IntegerPartitions[n-1]];
Table[Length[allbal[n]],{n,15}]

PartitionProduct(p,f)={my(r=1,k=0); for(i=1,length(p), if(i==length(p) || p[i]!=p[i+1], r*=f(p[i],i-k);k=i)); r}
UPick(total,kinds)=binomial(total+kinds-1,kinds-1);
D(n)={my(v=vector(n)); v[1]=[1]; for(n=2, n, v[n]=vector(n-1); forpart(p=n-1, for(leaves=1, length(v[p[1]]), v[n][leaves*length(p)]+=PartitionProduct(p,(t,e)->UPick(e,v[t][leaves]))))); v}
a(n)=vecsum(D(n)[n]); \\ Andrew Howroyd, Sep 02 2017

Terms a(26) and beyond from Andrew Howroyd, Sep 02 2017

A291441 Matula-Goebel numbers of orderless same-trees with all leaves equal to 1.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 256, 343, 361, 512, 1024, 2048, 2401, 2809, 4096, 6859, 8192, 12031, 16384, 16807, 17161, 32768, 51529, 65536, 96721, 117649, 130321, 131072, 148877, 262144, 516961, 524288, 637643, 718099, 757907, 823543, 1048576, 2097152, 2248091

Offset: 1

Gus Wiseman, Aug 23 2017

nonn

See A289078 for the definition of orderless same-tree.

			a(20)=12031 corresponds to the following same-tree: {{1,1,1,1},{{1,1},{1,1}}}.

Cf. A007097, A109129, A214577, A280994, A289078, A289079, A291442, A291443.

nn=200000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
leafcount[n_]:=If[n===1,1,With[{m=primeMS[n]},If[Length[m]===1,leafcount[First[m]],Total[leafcount/@m]]]];
sameQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]>1,SameQ@@leafcount/@m,And@@sameQ/@m]]];
Select[Range[nn],sameQ]

More terms from Jinyuan Wang, Jun 21 2020

A298120 Matula-Goebel numbers of rooted trees in which all positive outdegrees are odd.

Original entry on oeis.org

1, 2, 3, 5, 8, 11, 12, 18, 19, 20, 27, 30, 31, 32, 37, 44, 45, 48, 50, 61, 66, 67, 71, 72, 75, 76, 80, 99, 103, 108, 110, 113, 114, 120, 124, 125, 127, 128, 131, 148, 157, 162, 165, 171, 176, 180, 186, 190, 192, 193, 197, 200, 222, 223, 229, 242, 243, 244, 264

Offset: 1

Gus Wiseman, Jan 12 2018

nonn

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
5  (((o)))
8  (ooo)
11 ((((o))))
12 (oo(o))
18 (o(o)(o))
19 ((ooo))
20 (oo((o)))
27 ((o)(o)(o))
30 (o(o)((o)))
31 (((((o)))))
32 (ooooo)
37 ((oo(o)))
44 (oo(((o))))
45 ((o)(o)((o)))
48 (oooo(o))
50 (o((o))((o)))

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A007097, A026424, A061775, A214577, A276625, A277098, A290760, A291441, A291442, A291636, A297571, A298118.

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

A297571 Matula-Goebel numbers of fully unbalanced rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 13, 15, 22, 26, 29, 30, 31, 33, 39, 41, 47, 55, 58, 62, 65, 66, 78, 79, 82, 87, 93, 94, 101, 109, 110, 113, 123, 127, 130, 137, 141, 145, 155, 158, 165, 167, 174, 179, 186, 195, 202, 205, 211, 218, 226, 235, 237, 246, 254, 257, 271, 274

Offset: 1

Gus Wiseman, Dec 31 2017

nonn

An unlabeled rooted tree is fully unbalanced if either (1) it is a single node, or (2a) every branch has a different number of nodes and (2b) every branch is fully unbalanced also. The number of fully unbalanced trees with n nodes is A032305(n).

The first finitary number (A276625) not in this sequence is 143.

			Sequence of fully unbalanced trees 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)))))
  33 ((o)(((o))))
  39 ((o)(o(o)))
  41 (((o(o))))
  47 (((o)((o))))

Cf. A000081, A003238, A004111, A007097, A032305, A061775, A214577, A273873, A276625, A277098, A290689, A290760, A291441, A291442, A291443.

nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
MGweight[n_]:=If[n===1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>k*MGweight[PrimePi[p]]]]];
imbalQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[UnsameQ@@MGweight/@m,And@@imbalQ/@m]]];
Select[Range[nn],imbalQ]

A304173 Number of rooted plane trees where every branch that has a predecessor (a branch directly to its left and emanating from the same root) has at least as many leaves as its predecessor.

Original entry on oeis.org

1, 1, 2, 5, 13, 34, 90, 242, 660, 1822, 5085, 14333, 40759, 116817, 337140, 979098, 2859439, 8393113, 24747052, 73262246, 217681621, 648939319, 1940461444, 5818595438, 17492367097, 52712114792, 159193762250, 481754196170, 1460650624068, 4436422703787, 13496947320929

Offset: 1

Gus Wiseman, Aug 16 2018

nonn

			The a(5) = 13 plane trees:
  ((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
  (((o))o), (o((o))), (o(oo)), ((o)(o)),
  ((o)oo), (o(o)o), (oo(o)),
  (oooo).
Missing from this list is ((oo)o).

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

Cf. A000081, A000108, A001003, A001006, A003238, A007853, A126120, A291442, A291443, A304175.

pplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[pplane/@c],OrderedQ[Count[#,{},{0,Infinity}]&/@#]&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[pplane[n]],{n,10}]

seq(n)={my(p=x*y+O(x^2)); for(n=2, n, p=x*(y-1 + 1/prod(k=1, n-1, 1 - y^k*polcoef(p,k,y)))); Vec(subst(p,y,1))} \\ Andrew Howroyd, Jan 22 2021

G.f.: A(x,1) where A(x,y) satisfies A(x,y) = x*(y-1 + 1/(Product_{k>=1} 1 - y^k * [y^k] A(x,y))). - Andrew Howroyd, Jan 22 2021

Terms a(15) and beyond from Andrew Howroyd, Jan 22 2021

A304175 Number of leaf-balanced rooted plane trees with n nodes.

Original entry on oeis.org

1, 1, 2, 5, 12, 27, 59, 128, 277, 597, 1280, 2730, 5794, 12248, 25836, 54508, 115222, 244144, 518104, 1099499, 2330326, 4930089, 10415135, 21992400, 46470911, 98353146, 208580686, 443186181, 942988423, 2007981801, 4276830431, 9109431322, 19404918449, 41357252072, 88236092543

Offset: 1

Gus Wiseman, Aug 16 2018

nonn

A rooted plane tree is leaf-balanced if every branch of the root has the same number of leaves, and every branch of the root is itself leaf-balanced.

			The a(5) = 12 leaf-balanced plane trees:
  ((((o)))), (((oo))), (((o)o)), ((o(o))), ((ooo)),
  (((o))o), (o((o))), ((o)(o)),
  ((o)oo), (o(o)o), (oo(o)),
  (oooo).
Missing from this list are ((oo)o) and (o(oo)).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000081, A000108, A001003, A001006, A003238, A007853, A126120, A291442, A291443, A304173.

lbplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[lbplane/@c],SameQ@@(Count[#,{},{0,Infinity}]&/@#)&],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[lbplane[n]],{n,10}]

seq(n)={my(v=vector(n)); v[1]=x/(1-x) + O(x*x^n); for(k=2, n, v[k]=x*sumdiv(k, d, if(dAndrew Howroyd, Dec 13 2020

Terms a(17) and beyond from Andrew Howroyd, Dec 13 2020

A298536 Matula-Goebel numbers of rooted trees such that every branch of the root has a different number of leaves.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 34, 35, 37, 38, 39, 41, 43, 46, 47, 51, 53, 57, 58, 59, 61, 65, 67, 69, 71, 73, 74, 77, 79, 82, 83, 85, 86, 87, 89, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 115, 118, 122, 123, 127, 129, 131, 133

Offset: 1

Gus Wiseman, Jan 20 2018

nonn

			Sequence of trees begins:
1  o
2  (o)
3  ((o))
5  (((o)))
7  ((oo))
11 ((((o))))
13 ((o(o)))
14 (o(oo))
17 (((oo)))
19 ((ooo))
21 ((o)(oo))
23 (((o)(o)))
26 (o(o(o)))
29 ((o((o))))
31 (((((o)))))
34 (o((oo)))
35 (((o))(oo))
37 ((oo(o)))
38 (o(ooo))
39 ((o)(o(o)))
41 (((o(o))))
43 ((o(oo)))
46 (o((o)(o)))
47 (((o)((o))))

Cf. A000081, A007097, A061775, A111299, A214577, A276625, A290689, A290760, A291442, A298534, A298535.

nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
leafcount[n_]:=If[n===1,1,With[{m=primeMS[n]},If[Length[m]===1,leafcount[First[m]],Total[leafcount/@m]]]];
Select[Range[nn],UnsameQ@@leafcount/@primeMS[#]&]

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]

cp's OEIS Frontend

A291636 Matula-Goebel numbers of lone-child-avoiding rooted trees.

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A303431 Aperiodic tree numbers. Matula-Goebel numbers of aperiodic rooted trees.

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A291443 Number of leaf-balanced trees with n nodes.

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A291441 Matula-Goebel numbers of orderless same-trees with all leaves equal to 1.

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A298120 Matula-Goebel numbers of rooted trees in which all positive outdegrees are odd.

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A297571 Matula-Goebel numbers of fully unbalanced rooted trees.

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A304173 Number of rooted plane trees where every branch that has a predecessor (a branch directly to its left and emanating from the same root) has at least as many leaves as its predecessor.

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Mathematica

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A304175 Number of leaf-balanced rooted plane trees with n nodes.

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