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

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

Author

Gus Wiseman, Feb 04 2020

Keywords

• nonn

Comments

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]

Examples

			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}
		

Links

• Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

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.

Programs

Mathematica

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

Formula

Intersection of A291636 and A306202.