%I #9 Mar 07 2017 06:16:44

%S 2,5,6,12,13,14,15,17,22,24,28,30,31,35,37,38,39,43,44,45,47,48,51,55,

%T 56,58,60,67,69,70,76,78,79,82,88,89,90,91,93,94,95,96,102,105,106,

%U 107,109,110,111,112,113,116,117,118,119,120,129,135,138,140,142

%N Matula-Goebel numbers of bicentral rooted trees.

%C A tree is said to be bicentral if its center consists of two points (see the Harary reference, p. 35).

%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.

%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

%D A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.

%D J. M. Aldous and R. J. Wilson, Graphs and Applications, Springer, 2000 (p. 179).

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F 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 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.

%e 5 is in the sequence because the rooted tree with Matula-Goebel number 5 is the path-tree on 4 vertices which is bicentral.

%p 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: CC := {}: for n to 180 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) = true then CC := `union`(CC, {n}) else end if end do: CC;

%Y Cf. A198329, A198330.

%K nonn

%O 1,1

%A _Emeric Deutsch_, Nov 24 2011