A331937 -id:A331937 - cp's OEIS frontend

A331935 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees.

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 21, 24, 26, 27, 28, 32, 36, 38, 39, 42, 46, 48, 49, 52, 54, 56, 57, 63, 64, 69, 72, 74, 76, 78, 81, 84, 86, 91, 92, 96, 98, 104, 106, 108, 111, 112, 114, 117, 122, 126, 128, 129, 133, 138, 144, 146, 147, 148, 152, 156, 159

Offset: 1

Gus Wiseman, Feb 03 2020

nonn

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
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 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 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))
  21: ((o)(oo))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  39: ((o)(o(o)))
  42: (o(o)(oo))
The sequence of terms together with their prime indices begins:
    1: {}              46: {1,9}             98: {1,4,4}
    2: {1}             48: {1,1,1,1,2}      104: {1,1,1,6}
    4: {1,1}           49: {4,4}            106: {1,16}
    6: {1,2}           52: {1,1,6}          108: {1,1,2,2,2}
    8: {1,1,1}         54: {1,2,2,2}        111: {2,12}
    9: {2,2}           56: {1,1,1,4}        112: {1,1,1,1,4}
   12: {1,1,2}         57: {2,8}            114: {1,2,8}
   14: {1,4}           63: {2,2,4}          117: {2,2,6}
   16: {1,1,1,1}       64: {1,1,1,1,1,1}    122: {1,18}
   18: {1,2,2}         69: {2,9}            126: {1,2,2,4}
   21: {2,4}           72: {1,1,1,2,2}      128: {1,1,1,1,1,1,1}
   24: {1,1,1,2}       74: {1,12}           129: {2,14}
   26: {1,6}           76: {1,1,8}          133: {4,8}
   27: {2,2,2}         78: {1,2,6}          138: {1,2,9}
   28: {1,1,4}         81: {2,2,2,2}        144: {1,1,1,1,2,2}
   32: {1,1,1,1,1}     84: {1,1,2,4}        146: {1,21}
   36: {1,1,2,2}       86: {1,14}           147: {2,4,4}
   38: {1,8}           91: {4,6}            148: {1,1,12}
   39: {2,6}           92: {1,1,9}          152: {1,1,1,8}
   42: {1,2,4}         96: {1,1,1,1,1,2}    156: {1,1,2,6}

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 enumeration of these trees by leaves is A050381.
The locally disjoint version A331873.
The enumeration of these trees by nodes is A331934.
The case with at most one distinct non-leaf branch of any vertex is A331936.
Lone-child-avoiding rooted trees are counted by A001678.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Cf. A000081, A000669, A007097, A061775, A196050, A320269, A330951, A331871, A331872, A331874, A331937, A331965.

mseQ[n_]:=n==1||n==2||!PrimeQ[n]&&And@@mseQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],mseQ]

A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 133, 152, 172, 212, 214, 224, 256, 262, 266, 301, 304, 326, 344, 371, 424, 428, 448, 512, 524, 526, 532, 602, 608, 622, 652, 688, 742, 749, 766, 817, 848, 856, 886, 896, 917, 1007, 1024, 1048, 1052

Offset: 1

Gus Wiseman, Feb 04 2020

nonn

First differs from A331683 in having 133, the Matula-Goebel number of the tree ((oo)(ooo)).
Lone-child-avoiding means there are no unary branchings.
In a semi-identity tree, the non-leaf 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.
Consists of one, and all composite numbers that are n times a power of two, where n is a squarefree number whose prime indices already belong to the sequence, and a prime index of n is a number m such that prime(m) divides n. [Clarified by Peter Munn and Gus Wiseman, Jun 24 2021]

			The sequence of all lone-child-avoiding rooted semi-identity 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))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  133: ((oo)(ooo))
  152: (ooo(ooo))
  172: (oo(o(oo)))
  212: (oo(oooo))
  214: (o(oo(oo)))
The sequence of terms together with their prime indices begins:
    1: {}                 224: {1,1,1,1,1,4}
    4: {1,1}              256: {1,1,1,1,1,1,1,1}
    8: {1,1,1}            262: {1,32}
   14: {1,4}              266: {1,4,8}
   16: {1,1,1,1}          301: {4,14}
   28: {1,1,4}            304: {1,1,1,1,8}
   32: {1,1,1,1,1}        326: {1,38}
   38: {1,8}              344: {1,1,1,14}
   56: {1,1,1,4}          371: {4,16}
   64: {1,1,1,1,1,1}      424: {1,1,1,16}
   76: {1,1,8}            428: {1,1,28}
   86: {1,14}             448: {1,1,1,1,1,1,4}
  106: {1,16}             512: {1,1,1,1,1,1,1,1,1}
  112: {1,1,1,1,4}        524: {1,1,32}
  128: {1,1,1,1,1,1,1}    526: {1,56}
  133: {4,8}              532: {1,1,4,8}
  152: {1,1,1,8}          602: {1,4,14}
  172: {1,1,14}           608: {1,1,1,1,1,8}
  212: {1,1,16}           622: {1,64}
  214: {1,28}             652: {1,1,38}

The non-semi case is {1}.
Not requiring lone-child-avoidance gives A306202.
The locally disjoint version is A331683.
These trees are counted by A331966.
The semi-lone-child-avoiding case is A331994.
Matula-Goebel numbers of rooted identity trees are A276625.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Semi-identity trees are counted by A306200.
Cf. A001678, A004111, A007097, A061775, A122132, A196050, A300660, A316694, A320269, A331681, A331686, A331875, A331936, A331937, A331993.

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

Intersection of A291636 and A306202.

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]

A331963 Matula-Goebel numbers of semi-lone-child-avoiding rooted identity trees.

Original entry on oeis.org

1, 2, 6, 26, 39, 78, 202, 303, 334, 501, 606, 794, 1002, 1191, 1313, 2171, 2382, 2462, 2626, 3693, 3939, 3998, 4342, 4486, 5161, 5997, 6513, 6729, 7162, 7386, 7878, 8914, 10322, 10743, 11994, 12178, 13026, 13371, 13458, 15483, 15866, 16003, 16867, 18267, 19286

Offset: 1

Gus Wiseman, Feb 03 2020

nonn

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. It is an identity tree if the branches under 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.
Consists of one, two, and all nonprime squarefree numbers 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 identity trees together with their Matula-Goebel numbers begins:
    1: o
    2: (o)
    6: (o(o))
   26: (o(o(o)))
   39: ((o)(o(o)))
   78: (o(o)(o(o)))
  202: (o(o(o(o))))
  303: ((o)(o(o(o))))
  334: (o((o)(o(o))))
  501: ((o)((o)(o(o))))
  606: (o(o)(o(o(o))))
  794: (o(o(o)(o(o))))

Gus Wiseman, The first 63 semi-lone-child-avoiding rooted identity trees.

A subset of A276625 (MG-numbers of identity trees).
Not requiring an identity tree gives A331935.
The locally disjoint version is A331937.
These trees are counted by A331964.
The semi-identity case is A331994.
Matula-Goebel numbers of identity trees are A276625.
Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees are A331965.
Cf. A004111, A007097, A050381, A061775, A196050, A291636, A300660, A306202, A320269, A331681, A331873, A331875, A331933, A331934, A331936.

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

Intersection of A276625 (identity trees) and A331935 (semi-lone-child-avoiding).

A331964 Number of semi-lone-child-avoiding rooted identity trees with n vertices.

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 1, 2, 2, 4, 6, 10, 16, 27, 44, 74, 123, 209, 353, 602, 1026, 1760, 3019, 5203, 8977, 15538, 26930, 46792, 81415, 141939, 247795, 433307, 758672, 1330219, 2335086, 4104064, 7220937, 12718694, 22424283, 39574443, 69903759, 123584852, 218668323

Offset: 1

Gus Wiseman, Feb 04 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. It is an identity tree if the branches of any given vertex are all distinct.

			The a(9) = 2 through a(12) = 10 semi-lone-child-avoiding 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(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(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))))))
                                                       ((o)((o)(o(o(o)))))
                                                       ((o)(o((o)(o(o)))))
                                                       ((o(o))((o)(o(o))))
                                                       (o((o)((o)(o(o)))))

Andrew Howroyd, Table of n, a(n) for n = 1..1000
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.
Gus Wiseman, The semi-lone-child-avoiding rooted identity trees with up to 13 vertices.

The non-semi version is A000007.
Matula-Goebel numbers of these trees are A331963.
Rooted identity trees are A004111.
Semi-lone-child-avoiding rooted trees are A331934.
Cf. A000081, A001678, A050381, A276625, A300660, A306200, A316694, A331872, A331875, A331933, A331935, A331936, A331937, A331966.

ssei[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Select[Union[Sort/@Tuples[ssei/@c]],UnsameQ@@#&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[ssei[n]],{n,15}]

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(v=[1,1]); for(n=2, n-1, v=concat(v, WeighT(v)[n] - v[n])); v} \\ Andrew Howroyd, Feb 09 2020

Terms a(36) and beyond from Andrew Howroyd, Feb 09 2020

A352289 a(1) = 1 and thereafter a(n) = 2*prime(a(n-1)).

Original entry on oeis.org

1, 4, 14, 86, 886, 13766, 298154, 8455786, 300427382, 12942000398, 659492202274, 38995629272042, 2634767648759954, 200877694833442486, 17101872791349773894, 1611548301646589127698, 166820830882144877428234

Offset: 1

Kevin Ryde, Mar 16 2022

In Matula-Goebel tree codes, a(n) is a rooted caterpillar consisting of a path of n-1 internal vertices down, and n childless vertices under them so each has exactly 2 children.

Mir, Rosselló, and Rotger show that among phylogenic trees (meaning series-reduced, no vertex with just 1 child) with n childless vertices, tree a(n) has the largest total cophenetic index A352288(a(n)) = binomial(n,3).

			For n=3, a(3) = 14 is the Matula-Goebel code of the following tree
  root  14
       /  \     tree numbers of subtrees shown,
      4    1    with "1" being childless,
     / \        and n=3 of those
    1   1

Lucas A. Brown, Python program.
Arnau Mir, Francesc Rosselló, and Lucía Rotger, A New Balance Index for Phylogenetic Trees, arXiv:1202.1223 [q-bio.PE], 2012.

Cf. A352288 (total cophenetic index).

Cf. A331937.

NestList[2 Prime[#] &, 1, 10] (* Michael De Vlieger, Apr 18 2022 *)

a(n) = my(ret=1); for(i=2,n, ret=2*prime(ret)); ret;

from functools import lru_cache
from sympy import prime
@lru_cache(maxsize=None)
def A352289(n): return 1 if n == 1 else 2*prime(A352289(n-1)) # Chai Wah Wu, Apr 18 2022

a(10)-a(15) from Daniel Suteu, Mar 19 2022

a(16) and a(17) from Lucas A. Brown, Sep 04 2024

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A331935 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees.

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A331965 Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.

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

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A331963 Matula-Goebel numbers of semi-lone-child-avoiding rooted identity trees.

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A331964 Number of semi-lone-child-avoiding rooted identity trees with n vertices.

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A352289 a(1) = 1 and thereafter a(n) = 2*prime(a(n-1)).

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