OFFSET
1,1
COMMENTS
We say that a rooted tree is (topologically) series-reduced if no vertex has degree 2.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
Also Matula-Goebel numbers of lone-child-avoiding rooted trees with more than two branches.
LINKS
David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014)
Eric Weisstein's World of Mathematics, Series-reduced tree.
EXAMPLE
The sequence of all series-reduced rooted trees with more than two branches together with their Matula-Goebel numbers begins:
8: (ooo)
16: (oooo)
28: (oo(oo))
32: (ooooo)
56: (ooo(oo))
64: (oooooo)
76: (oo(ooo))
98: (o(oo)(oo))
112: (oooo(oo))
128: (ooooooo)
152: (ooo(ooo))
172: (oo(o(oo)))
196: (oo(oo)(oo))
212: (oo(oooo))
224: (ooooo(oo))
256: (oooooooo)
266: (o(oo)(ooo))
304: (oooo(ooo))
343: ((oo)(oo)(oo))
344: (ooo(o(oo)))
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
srQ[n_]:=Or[n==1, With[{m=primeMS[n]}, And[Length[m]>1, And@@srQ/@m]]];
Select[Range[1000], PrimeOmega[#]>2&&srQ[#]&]
CROSSREFS
These trees are counted by A331488.
Unlabeled rooted trees are counted by A000081.
Lone-child-avoiding rooted trees are counted by A001678.
Topologically series-reduced rooted trees are counted by A001679.
Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636.
Matula-Goebel numbers of series-reduced rooted trees are A331489.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 20 2020
STATUS
approved