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A339193
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Matula-Goebel numbers of unlabeled binary rooted semi-identity trees.
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5
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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
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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))))
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MATHEMATICA
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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]
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CROSSREFS
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Counting these trees by number of nodes gives A063895.
A000081 counts unlabeled rooted trees with n nodes.
A331965 ranks lone-child avoiding semi-identity trees, counted by A331966.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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