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A184162
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Number of chains in the rooted tree with Matula-Goebel number n.
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1
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1, 3, 7, 5, 15, 9, 11, 7, 13, 17, 31, 11, 19, 13, 21, 9, 23, 15, 15, 19, 17, 33, 27, 13, 29, 21, 19, 15, 35, 23, 63, 11, 37, 25, 25, 17, 23, 17, 25, 21, 39, 19, 27, 35, 27, 29, 43, 15, 21, 31, 29, 23, 19, 21, 45, 17, 21, 37, 47, 25, 31, 65, 23, 13, 33, 39, 31, 27, 33, 27, 39, 19, 35, 25, 35, 19, 41, 27, 67, 23
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OFFSET
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1,2
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COMMENTS
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The vertices of a rooted tree can be regarded as a partially ordered set, where u<=v holds for two vertices u and v if and only if u lies on the unique path between v and the root. A chain is a nonempty set of pairwise comparable vertices.
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)=1; if n=p(t) (=the t-th prime), then a(n)=1+2a(t); if n=rs (r,s,>=2), then a(n)=a(r)+a(s)-1. The Maple program is based on this recursive formula.
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EXAMPLE
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a(5) = 15 because the rooted tree with Matula-Goebel number 5 is a path ABCD on 4 vertices and any nonempty subset of {A,B,C,D} is a chain.
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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 1 elif bigomega(n) = 1 then 1+2*a(pi(n)) else a(r(n))+a(s(n))-1 end if end proc: seq(a(n), n = 1 .. 80);
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PROG
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(Haskell)
import Data.List (genericIndex)
a184162 n = genericIndex a184162_list (n - 1)
a184162_list = 1 : g 2 where
g x = y : g (x + 1) where
y = if t > 0 then 2 * a184162 t + 1 else a184162 r + a184162 s - 1
where t = a049084 x; r = a020639 x; s = x `div` r
(PARI) a(n) = my(f=factor(n)); [self()(primepi(p)) |p<-f[, 1]] * f[, 2]*2 + 1; \\ Kevin Ryde, Aug 25 2021
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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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