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A196057
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Number of sibling pairs in the rooted tree with Matula-Goebel number n.
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1
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0, 0, 0, 1, 0, 1, 1, 3, 1, 1, 0, 3, 1, 2, 1, 6, 1, 3, 3, 3, 2, 1, 1, 6, 1, 2, 3, 4, 1, 3, 0, 10, 1, 2, 2, 6, 3, 4, 2, 6, 1, 4, 2, 3, 3, 2, 1, 10, 3, 3, 2, 4, 6, 6, 1, 7, 4, 2, 1, 6, 3, 1, 4, 15, 2, 3, 3, 4, 2, 4, 3, 10, 2, 4, 3, 6, 2, 4, 1, 10, 6, 2, 1, 7, 2, 3, 2, 6, 6, 6, 3, 4, 1, 2, 4, 15, 1, 5, 3, 6, 2, 4, 3, 7, 4, 7, 4, 10, 1, 3
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OFFSET
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1,8
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COMMENTS
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A sibling pair is an unordered pair of nodes having the same parent.
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; if n=p(t) (=the t-th prime), then a(n)=a(t); if n=rs (r,s,>=2), then a(n) = a(r) + a(s) + G(r)G(s), where G(m) is 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)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = binomial(m,2) 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 u, v: u := proc (n) options operator, arrow: op(1, factorset(n)) end proc: v := proc (n) options operator, arrow: n/u(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n)) else a(u(n))+a(v(n))+bigomega(u(n))*bigomega(v(n)) end if end proc: seq(a(n), n = 1 .. 110);
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PROG
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(Haskell)
import Data.List (genericIndex)
a196057 n = genericIndex a196057_list (n - 1)
a196057_list = 0 : g 2 where
g x = y : g (x + 1) where
y | t > 0 = a196057 t
| otherwise = a196057 r + a196057 s + 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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