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
Keywords
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
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.
Comments