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A214567
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Maximal number of distinct rooted trees obtained from the rooted tree with Matula-Goebel number n by adding one pendant edge at one of its vertices.
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3
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1, 2, 3, 2, 4, 4, 3, 2, 3, 5, 5, 4, 5, 4, 6, 2, 4, 4, 3, 5, 5, 6, 4, 4, 4, 6, 3, 4, 6, 7, 6, 2, 7, 5, 6, 4, 5, 4, 7, 5, 6, 6, 5, 6, 6, 5, 7, 4, 3, 5, 6, 6, 3, 4, 8, 4, 5, 7, 5, 7, 5, 7, 5, 2, 8, 8, 4, 5, 6, 7, 6, 4, 6, 6, 6, 4, 7, 8, 7, 5, 3
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OFFSET
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1,2
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COMMENTS
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The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
Sum_{j such that rooted tree with Matula-Goebel number j has n vertices} a(j) = A000107(n). Example: the Matula-Goebel numbers of the rooted trees with 4 vertices are 5,6,7,8 and a(5)+a(6)+a(7)+a(8) = 4+4+3+2=13 = A000107(4).
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LINKS
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FORMULA
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a(1)=1; if n is t-th prime, then a(n)=1+a(t); if n is composite, then a(n) = 1+Sum_{p|n}(a(p)-1), where summation is over the distinct prime divisors of n.
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EXAMPLE
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a(4)=2 because the rooted tree with Matula-Goebel number 4 is V; adding an edge at either of the two leaves yields the same rooted tree.
a(5)=4 because the rooted tree with Matula-Goebel number 5 is the path on 4 vertices; adding one edge at any of the vertices yields a new rooted tree.
a(987654321)=18 (reader may verify this on Fig. 2 of the Deutsch paper).
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MAPLE
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with(numtheory): a := proc (n) local FS: FS := proc (n) options operator, arrow: factorset(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+a(pi(n)) else 1+add(a(FS(n)[j])-1, j = 1 .. nops(FS(n))) end if end proc: seq(a(n), n = 1 .. 130);
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PROG
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(Haskell)
import Data.List (genericIndex)
a214567 n = genericIndex a214567_list (n - 1)
a214567_list = 1 : g 2 where
g x = y : g (x + 1) where
y | t > 0 = a214567 t + 1
| otherwise = 1 + sum (map ((subtract 1) . a214567) $ a027748_row x)
where t = a049084 x
(PARI) a(n) = 1 + vecsum([self()(primepi(p)) |p<-factor(n)[, 1]]); \\ Kevin Ryde, Oct 19 2022
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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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