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A331963
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Matula-Goebel numbers of semi-lone-child-avoiding rooted identity trees.
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12
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1, 2, 6, 26, 39, 78, 202, 303, 334, 501, 606, 794, 1002, 1191, 1313, 2171, 2382, 2462, 2626, 3693, 3939, 3998, 4342, 4486, 5161, 5997, 6513, 6729, 7162, 7386, 7878, 8914, 10322, 10743, 11994, 12178, 13026, 13371, 13458, 15483, 15866, 16003, 16867, 18267, 19286
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OFFSET
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1,2
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COMMENTS
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A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. It is an identity tree if the branches under any given vertex are all distinct.
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.
Consists of one, two, and all nonprime squarefree numbers whose prime indices already belong to the sequence, where a prime index of n is a number m such that prime(m) divides n.
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LINKS
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FORMULA
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Intersection of A276625 (identity trees) and A331935 (semi-lone-child-avoiding).
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EXAMPLE
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The sequence of all semi-lone-child-avoiding rooted identity trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
6: (o(o))
26: (o(o(o)))
39: ((o)(o(o)))
78: (o(o)(o(o)))
202: (o(o(o(o))))
303: ((o)(o(o(o))))
334: (o((o)(o(o))))
501: ((o)((o)(o(o))))
606: (o(o)(o(o(o))))
794: (o(o(o)(o(o))))
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MATHEMATICA
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msiQ[n_]:=n==1||n==2||!PrimeQ[n]&&SquareFreeQ[n]&&And@@msiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[1000], msiQ]
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CROSSREFS
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A subset of A276625 (MG-numbers of identity trees).
Not requiring an identity tree gives A331935.
The locally disjoint version is A331937.
These trees are counted by A331964.
Matula-Goebel numbers of identity trees are A276625.
Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees are A331965.
Cf. A004111, A007097, A050381, A061775, A196050, A291636, A300660, A306202, A320269, A331681, A331873, A331875, A331933, A331934, A331936.
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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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