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

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

Author

Gus Wiseman, Feb 03 2020

Keywords

• nonn

Comments

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.

Examples

			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))))
		

Links

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

Crossrefs

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.

Programs

Mathematica

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

Formula

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