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A320269
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Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).
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11
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1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052
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OFFSET
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1,2
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COMMENTS
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First differs from A331871 in lacking 1589.
Lone-child-avoiding means there are no unary branchings.
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.
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LINKS
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EXAMPLE
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The sequence of rooted trees together with their Matula-Goebel numbers begins:
1: o
4: (oo)
8: (ooo)
14: (o(oo))
16: (oooo)
28: (oo(oo))
32: (ooooo)
38: (o(ooo))
49: ((oo)(oo))
56: (ooo(oo))
64: (oooooo)
76: (oo(ooo))
86: (o(o(oo)))
98: (o(oo)(oo))
106: (o(oooo))
112: (oooo(oo))
128: (ooooooo)
152: (ooo(ooo))
172: (oo(o(oo)))
196: (oo(oo)(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}]]]]
hmakQ[n_]:=And[!PrimeQ[n], SameQ@@DeleteCases[primeMS[n], 1], And@@hmakQ/@primeMS[n]]; Select[Range[1000], hmakQ[#]&]
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CROSSREFS
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Not requiring lone-child-avoidance gives A320230.
The enumeration of these trees by vertices is A320268.
The semi-lone-child-avoiding version is A331936.
If the non-leaf branches are all different instead of equal we get A331965.
Achiral rooted trees are counted by A003238.
MG-numbers of lone-child-avoiding rooted trees are A291636.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Updated with corrected terminology by Gus Wiseman, Feb 06 2020
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STATUS
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approved
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