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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..59.

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

Formula

Intersection of A306202 and A331935.

Example

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}

Mathematica

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

Crossrefs

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.

Sequence in context: A334267 A163823 A015929 * A043723 A376508 A331681

Adjacent sequences: A331991 A331992 A331993 * A331995 A331996 A331997

Keyword

nonn

Author

Gus Wiseman, Feb 05 2020

Status

approved