

A214577


The MatulaGoebel numbers of the generalized Bethe trees. A generalized Bethe tree is a rooted tree in which vertices at the same level have the same degree.


51



2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 27, 31, 32, 49, 53, 59, 64, 67, 81, 83, 97, 103, 121, 125, 127, 128, 131, 227, 241, 243, 256, 277, 289, 311, 331, 343, 361, 419, 431, 509, 512, 529, 563, 625, 661, 691, 709, 719, 729, 739, 961, 1024, 1331, 1433, 1523, 1543, 1619, 1787, 1879, 2048, 2063, 2187, 2221, 2309, 2401, 2437, 2809, 2897
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OFFSET

1,1


COMMENTS

The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.
Generalized Bethe trees are called uniform trees in the Goldberg  Livshits reference.
There is a simple bijection between generalized Bethe trees with n edges and partitions of n in which each part is divisible by the next (the parts are given by the number of edges at the successive levels). We have the correspondences: number of edges  sum of parts; root degree  last part; number of leaves  first part; height  number of parts.


LINKS

Table of n, a(n) for n=1..71.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 23142322.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
M. K. Goldberg and E. M. Livshits, On minimal universal trees, Mathematical Notes of the Acad. of Sciences of the USSR, 4, 1968, 713717 (translation from the Russian Mat. Zametki 4 1968 371379).
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
O. Rojo, Spectra of weighted generalized Bethe trees joined at the root, Linear Algebra and its Appl., 428, 2008, 29612979.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

In A214578 one has defined Q(n)=0 if n is the MatulaGoebel number of a rooted tree that is not a generalized Bethe tree and Q(n) to be a certain polynomial if n corresponds to a generalized Bethe tree. The Maple program makes use of this to find the MatulaGoebel numbers corresponding to the generalized Bethe trees.


EXAMPLE

7 is in the sequence because the corresponding rooted tree is Y, a generalized Bethe tree.


MAPLE

with(numtheory): Q := 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 and Q(pi(n)) = 0 then 0 elif bigomega(n) = 1 then sort(expand(1+x*Q(pi(n)))) elif Q(r(n)) <> 0 and Q(s(n)) <> 0 and type(simplify(Q(r(n))/Q(s(n))), constant) = true then sort(Q(r(n))+Q(s(n))) else 0 end if end proc: A := {}; for n to 3000 do if Q(n) = 0 then else A := `union`(A, {n}) end if end do: A;


CROSSREFS

Cf. A214578.
Differs from A243497 for the first time at n=31.
Sequence in context: A318690 A302498 A243497 * A280994 A138039 A289995
Adjacent sequences: A214574 A214575 A214576 * A214578 A214579 A214580


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Aug 18 2012


STATUS

approved



