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A198329
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The Matula-Goebel number of the rooted tree obtained from the rooted tree with Matula-Goebel number n after removing the vertices of degree one, together with their incident edges.
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5
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1, 1, 1, 1, 2, 2, 1, 1, 4, 3, 3, 2, 2, 2, 6, 1, 2, 4, 1, 3, 4, 5, 4, 2, 9, 3, 8, 2, 3, 6, 5, 1, 10, 3, 6, 4, 2, 2, 6, 3, 3, 4, 2, 5, 12, 7, 6, 2, 4, 9, 6, 3, 1, 8, 15, 2, 4, 5, 3, 6, 4, 11, 8, 1, 9, 10, 2, 3, 14, 6, 3, 4, 4, 3, 18, 2, 10, 6, 5, 3, 16, 5, 7
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OFFSET
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1,5
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COMMENTS
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This is the pruning operation mentioned, for example, in the Balaban reference (p. 360) and in the Todeschini - Consonni reference (p. 42).
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REFERENCES
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A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.
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LINKS
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FORMULA
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Let b(n)=A198328(n) (= the Matula-Goebel number of the tree obtained by removing the leaves together with their incident edges from the rooted tree with Matula-Goebel number n). a(1)=1; if n = p(t) (=the t-th prime), then a(n)=b(t); if n=rs (r,s>=2), then a(n)=b(r)b(s). The Maple program is based on this recursive formula.
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EXAMPLE
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a(7)=1 because the rooted tree with Matula-Goebel number 7 is Y; after deleting the vertices of degree one and their incident edges we obtain the 1-vertex tree having Matula-Goebel number 1.
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MAPLE
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with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: seq(a(n), n = 1 .. 120);
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MATHEMATICA
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b[1] = b[2] = 1; b[n_] := b[n] = If[PrimeQ[n], Prime[b[PrimePi[n]]], Times @@ (b[#[[1]]]^#[[2]]& /@ FactorInteger[n])];
a[1] = 1; a[n_] := a[n] = If[PrimeQ[n], b[PrimePi[n]], Times @@ (b[#[[1]] ]^#[[2]]& /@ FactorInteger[n])];
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CROSSREFS
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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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