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 A198330 Matula-Goebel numbers of central rooted trees. 1
 1, 3, 4, 7, 8, 9, 10, 11, 16, 18, 19, 20, 21, 23, 25, 26, 27, 29, 32, 33, 34, 36, 40, 41, 42, 46, 49, 50, 52, 53, 54, 57, 59, 61, 62, 63, 64, 65, 66, 68, 71, 72, 73, 74, 75, 77, 80, 81, 83, 84, 85, 86, 87, 92, 97, 98, 99, 100, 101, 103, 104, 108, 114, 115 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS A tree is said to be central if its center consists of one point (see the Harary reference, p. 35). REFERENCES F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143. I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142. I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22. D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273. A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979. J. M. Aldous and R. J. Wilson, Graphs and Applications, Springer, 2000 (p. 179). LINKS FORMULA A198329(n) is the Matula-Goebel number of the rooted  tree obtained by removing from the rooted tree with Matula-Goebel number n the vertices of degree one, together with their incident edges. If the repeated application of this pruning operation will not lead to the Matula-Goebel number 2 (corresponding to the 1-edge tree), then the starting rooted tree is central. The Maple program is based on this. EXAMPLE 7 is in the sequence because the rooted tree with Matula-Goebel number 7 is Y which is central. 5 is not in the sequence because the corresponding rooted tree is the path-tree on 4 vertices, a bicentral tree. MAPLE 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: C := {}: for n to 130 do B := {n}: nn := n: for i while a(nn) > 1 do B := `union`(B, {a(nn)}): nn := a(nn) end do: if member(2, B) = false then C := `union`(C, {n}) else  end if end do: C; CROSSREFS Cf. A198329, A198331. Sequence in context: A193406 A104426 A097044 * A232779 A086986 A057811 Adjacent sequences:  A198327 A198328 A198329 * A198331 A198332 A198333 KEYWORD nonn AUTHOR Emeric Deutsch, Nov 24 2011 STATUS approved

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Last modified January 20 14:40 EST 2019. Contains 319333 sequences. (Running on oeis4.)