

A339193


MatulaGoebel numbers of unlabeled binary rooted semiidentity trees.


5



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

1,2


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 semiidentity tree, only the nonleaf branches of any given vertex are distinct. Alternatively, a rooted tree is a semiidentity tree if the nonleaf branches of the root are all distinct and are themselves semiidentity trees.
The MatulaGoebel number of an unlabeled rooted tree is the product of primes indexed by the MatulaGoebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.


LINKS

Table of n, a(n) for n=1..22.
Gus Wiseman, The sequence of all unlabeled binary rooted semiidentity trees by MatulaGoebel number.


EXAMPLE

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))))


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]


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 semiidentity trees, counted by A306200.
A306203 ranks balanced semiidentity trees, counted by A306201.
A331965 ranks lonechild avoiding semiidentity trees, counted by A331966.
Cf. A007097, A061775, A196050, A291636, A331963, A331964.
Sequence in context: A327355 A024421 A259353 * A352289 A330465 A202139
Adjacent sequences: A339190 A339191 A339192 * A339194 A339195 A339196


KEYWORD

nonn


AUTHOR

Gus Wiseman, Mar 14 2021


STATUS

approved



