A331936 -id:A331936 - cp's OEIS frontend

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

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.

A331991 Number of semi-lone-child-avoiding achiral rooted trees with n vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 4, 4, 1, 7, 1, 7, 5, 6, 1, 7, 3, 7, 5, 7, 1, 13, 1, 8, 6, 6, 6, 10, 1, 9, 7, 9, 1, 15, 1, 12, 12, 8, 1, 12, 4, 13, 6, 11, 1, 15, 7, 13, 9, 9, 1, 17, 1, 15, 15, 9, 8, 21, 1, 13, 8, 16, 1, 18, 1, 12, 16, 11, 8, 21, 1

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 a(n) trees for n = 2, 3, 5, 7, 11, 13:
  (o)  (oo)  (oooo)    (oooooo)     (oooooooooo)        (oooooooooooo)
             ((o)(o))  ((oo)(oo))   ((oooo)(oooo))      ((ooooo)(ooooo))
                       ((o)(o)(o))  ((o)(o)(o)(o)(o))   ((ooo)(ooo)(ooo))
                                    (((o)(o))((o)(o)))  ((oo)(oo)(oo)(oo))
                                                        ((o)(o)(o)(o)(o)(o))

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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.

Matula-Goebel numbers of these trees are A331992.
The fully lone-child-avoiding case is A167865.
The semi-achiral version is A331933.
Not requiring achirality gives A331934.
The identity tree version is A331964.
The semi-identity tree version is A331993.
Achiral rooted trees are counted by A003238.
Lone-child-avoiding semi-achiral trees are A320268.
Cf. A000081, A004111, A050381, A001678, A214577, A289079, A320222, A331912, A331935, A331936, A331963, A331967, A331994.

ab[n_]:=If[n<=2,1,Sum[ab[d],{d,Most[Divisors[n-1]]}]];
Array[ab,100]

a(1) = a(2) = 1; a(n + 1) = Sum_{d|n, d 1.

G.f. A(x) satisfies: A(x) = x * (1 + (1/(1 + x)) * Sum_{k>=1} A(x^k)). - Ilya Gutkovskiy, Feb 25 2020

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).

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A331991 Number of semi-lone-child-avoiding achiral rooted trees with n vertices.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula