A214577 The Matula-Goebel 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.

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

Offset: 1

Author

Emeric Deutsch, Aug 18 2012

Keywords

• nonn

Comments

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.

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.

Examples

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

Links

• Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
• Emeric Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
• F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
• M. K. Goldberg and E. M. Livshits, On minimal universal trees, Mathematical Notes of the Acad. of Sciences of the USSR, 4, 1968, 713-717 (translation from the Russian Mat. Zametki 4 1968 371-379).
• 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 Rev. 10 (1968) 273.
• O. Rojo, Spectra of weighted generalized Bethe trees joined at the root, Linear Algebra and its Appl., 428, 2008, 2961-2979.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A214578.

Differs from A243497 for the first time at n=31.

Programs

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;

Mathematica

r[n_Integer] := r[n] = FactorInteger[n][[1, 1]];
s[n_Integer] := n/r[n];
Q[n_Integer] := Cancel@ Together@ Simplify@ Which[n == 1, 0, n == 2, 1, PrimeOmega[n] == 1 && Q[PrimePi[n]] === 0, 0, PrimeOmega[n] == 1, 1 + x * Q[PrimePi[n]], Q[r[n]] =!= 0 && Q[s[n]] =!= 0 && FreeQ[Q[r[n]]/Q[s[n]], x], Q[r[n]] + Q[s[n]], True, 0];
A = {};
For[n = 1, n <= 3000, n++, If[Q[n] === 0, , Print[n, " ", Q[n]]; A = Union[A, {n}]]];
A (* Jean-François Alcover, Aug 03 2024, after Emeric Deutsch *)

Formula

In A214578 one has defined Q(n)=0 if n is the Matula-Goebel 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 Matula-Goebel numbers corresponding to the generalized Bethe trees.