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A196063
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The Narumi-Katayama index of the rooted tree with Matula-Goebel number n.
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2
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0, 1, 2, 2, 4, 4, 3, 3, 8, 8, 8, 6, 6, 6, 16, 4, 6, 12, 4, 12, 12, 16, 12, 8, 32, 12, 24, 9, 12, 24, 16, 5, 32, 12, 24, 16, 8, 8, 24, 16, 12, 18, 9, 24, 48, 24, 24, 10, 18, 48, 24, 18, 5, 32, 64, 12, 16, 24, 12, 32, 16, 32, 36, 6, 48, 48, 8, 18, 48, 36, 16, 20, 18, 16, 96, 12, 48, 36, 24, 20, 64, 24, 24, 24, 48, 18, 48, 32, 10, 64
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OFFSET
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1,3
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COMMENTS
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The Narumi-Katayama index of a connected graph is the product of the degrees of the vertices of the graph.
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.
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LINKS
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FORMULA
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a(1)=0; a(2)=1; if n = prime(t) (the t-th prime, t>=2), then a(n)=a(t)*(1+G(t))/G(t); if n=rs (r,s>=2), then a(n)=a(r)*a(s)*G(n)/[G(r)*G(s)]; G(m) denotes the number of prime divisors of m counted with multiplicities. The Maple program is based on this recursive formula.
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EXAMPLE
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a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*3*1*1=3).
a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
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MAPLE
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with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n))*(1+bigomega(pi(n)))/bigomega(pi(n)) else a(r(n))*a(s(n))*bigomega(n)/(bigomega(r(n))*bigomega(s(n))) end if end proc: seq(a(n), n = 1 .. 90);
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PROG
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(Haskell)
import Data.List (genericIndex)
a196063 n = genericIndex a196063_list (n - 1)
a196063_list = 0 : 1 : g 3 where
g x = y : g (x + 1) where
y | t > 0 = a196063 t * (a001222 t + 1) `div` a001222 t
| otherwise = a196063 r * a196063 s * a001222 x `div`
(a001222 r * a001222 s)
where t = a049084 x; r = a020639 x; s = x `div` r
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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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