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A330943
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Matula-Goebel numbers of singleton-reduced rooted trees.
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10
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1, 2, 4, 6, 7, 8, 12, 13, 14, 16, 18, 19, 21, 24, 26, 28, 32, 34, 36, 37, 38, 39, 42, 43, 48, 49, 52, 53, 54, 56, 57, 61, 63, 64, 68, 72, 73, 74, 76, 78, 82, 84, 86, 89, 91, 96, 98, 101, 102, 104, 106, 107, 108, 111, 112, 114, 117, 119, 122, 126, 128, 129, 131
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OFFSET
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1,2
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COMMENTS
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These trees are counted by A330951.
A rooted tree is singleton-reduced if no non-leaf node has all singleton branches, where a rooted tree is a singleton if its root has degree 1.
The Matula-Goebel number of a rooted tree is the product of primes of the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
A prime index of n is a number m such that prime(m) divides n. A number belongs to this sequence iff it is 1 or its prime indices all belong to this sequence but are not all prime.
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LINKS
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Table of n, a(n) for n=1..63.
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EXAMPLE
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The sequence of all singleton-reduced rooted trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
4: (oo)
6: (o(o))
7: ((oo))
8: (ooo)
12: (oo(o))
13: ((o(o)))
14: (o(oo))
16: (oooo)
18: (o(o)(o))
19: ((ooo))
21: ((o)(oo))
24: (ooo(o))
26: (o(o(o)))
28: (oo(oo))
32: (ooooo)
34: (o((oo)))
36: (oo(o)(o))
37: ((oo(o)))
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
mgsingQ[n_]:=n==1||And@@mgsingQ/@primeMS[n]&&!And@@PrimeQ/@primeMS[n];
Select[Range[100], mgsingQ]
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CROSSREFS
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The series-reduced case is A291636.
Unlabeled rooted trees are counted by A000081.
Numbers whose prime indices are not all prime are A330945.
Singleton-reduced rooted trees are counted by A330951.
Singleton-reduced phylogenetic trees are A000311.
The set S of numbers whose prime indices do not all belong to S is A324694.
Cf. A000669, A001678, A006450, A007097, A007821, A061775, A196050, A257994, A276625, A277098, A320628, A330944, A330948.
Sequence in context: A228370 A186112 A029453 * A352089 A358458 A014855
Adjacent sequences: A330940 A330941 A330942 * A330944 A330945 A330946
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KEYWORD
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nonn
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AUTHOR
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Gus Wiseman, Jan 13 2020
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STATUS
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approved
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