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A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.

Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of the its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Consists of one, two, and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.

allocated for Gus WisemanMatula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 69, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 138, 144, 148, 152, 161, 162, 169, 172, 178, 184, 192, 196, 202, 206, 207, 208, 212, 214, 216, 224, 243

1,2

First differs from A331936 in having 69, the Matula-Goebel number of the tree ((o)((o)(o))).

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of the root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Consists of one, two, and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime.

David Callan, <a href="http://arxiv.org/abs/1406.7784">A sign-reversing involution to count labeled lone-child-avoiding trees</a>, arXiv:1406.7784 [math.CO], (30-June-2014).

Gus Wiseman, <a href="https://docs.google.com/document/d/e/2PACX-1vS1zCO9fgAIe5rGiAhTtlrOTuqsmuPos2zkeFPYB80gNzLb44ufqIqksTB4uM9SIpwlvo-oOHhepywy/pub">Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.</a>

The sequence of all semi-lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:

1: o

2: (o)

4: (oo)

6: (o(o))

8: (ooo)

9: ((o)(o))

12: (oo(o))

14: (o(oo))

16: (oooo)

18: (o(o)(o))

24: (ooo(o))

26: (o(o(o)))

27: ((o)(o)(o))

28: (oo(oo))

32: (ooooo)

36: (oo(o)(o))

38: (o(ooo))

46: (o((o)(o)))

48: (oooo(o))

49: ((oo)(oo))

msQ[n_]:=n==1||n==2||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];

Select[Range[100], msQ]

Not requiring lone-child-avoidance gives A316495.

A superset of A320269.

The semi-identity tree case is A331681.

The non-semi version (i.e., not containing 2) is A331871.

These trees counted by vertices are A331872.

These trees counted by leaves are A331874.

Not requiring local disjointness gives A331935.

The identity tree case is A331937.

Cf. A007097, A050381, A061775, A196050, A291636, A302696, A316473, A316696, A316697, A331680, A331682, A331683, A331687, A331934.

allocated

nonn

Gus Wiseman, Feb 02 2020

approved

editing

allocated for Gus Wiseman

allocated

approved