A339193 Matula-Goebel numbers of unlabeled binary rooted semi-identity trees.

1, 4, 14, 86, 301, 886, 3101, 3986, 13766, 13951, 19049, 48181, 57026, 75266, 85699, 199591, 263431, 295969, 298154, 302426, 426058, 882899

Offset: 1

Author

Gus Wiseman, Mar 14 2021

Keywords

• nonn

Comments

Definition: A positive integer belongs to the sequence iff it is 1, 4, or a squarefree semiprime whose prime indices both already belong to the sequence. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

In a semi-identity tree, only the non-leaf branches of any given vertex are distinct. Alternatively, a rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees.

The Matula-Goebel number of an unlabeled 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.

Examples

			The sequence of terms together with the corresponding unlabeled rooted trees begins:
      1: o
      4: (oo)
     14: (o(oo))
     86: (o(o(oo)))
    301: ((oo)(o(oo)))
    886: (o(o(o(oo))))
   3101: ((oo)(o(o(oo))))
   3986: (o((oo)(o(oo))))
  13766: (o(o(o(o(oo)))))
  13951: ((oo)((oo)(o(oo))))
  19049: ((o(oo))(o(o(oo))))
  48181: ((oo)(o(o(o(oo)))))
  57026: (o((oo)(o(o(oo)))))
  75266: (o(o((oo)(o(oo)))))
  85699: ((o(oo))((oo)(o(oo))))
		

Links

• Gus Wiseman, The sequence of all unlabeled binary rooted semi-identity trees by Matula-Goebel number.

Crossrefs

Counting these trees by number of nodes gives A063895.

A000081 counts unlabeled rooted trees with n nodes.

A111299 ranks binary trees, counted by A001190.

A276625 ranks identity trees, counted by A004111.

A306202 ranks semi-identity trees, counted by A306200.

A306203 ranks balanced semi-identity trees, counted by A306201.

A331965 ranks lone-child avoiding semi-identity trees, counted by A331966.

Cf. A007097, A061775, A196050, A291636, A331963, A331964.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mgbiQ[n_]:=Or[n==1,n==4,SquareFreeQ[n]&&PrimeOmega[n]==2&&And@@mgbiQ/@primeMS[n]];
Select[Range[1000],mgbiQ]