%I #18 Aug 03 2024 11:17:23
%S 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,
%T 121,125,127,128,131,227,241,243,256,277,289,311,331,343,361,419,431,
%U 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
%N 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.
%C 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.
%C Generalized Bethe trees are called uniform trees in the Goldberg - Livshits reference.
%C 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.
%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H Emeric Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.
%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H M. K. Goldberg and E. M. Livshits, <a href="http://mi.mathnet.ru/eng/mz9458">On minimal universal trees</a>, Mathematical Notes of the Acad. of Sciences of the USSR, 4, 1968, 713-717 (translation from the Russian Mat. Zametki 4 1968 371-379).
%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H O. Rojo, <a href="http://dx.doi.org/10.1016/j.laa.2008.01.026">Spectra of weighted generalized Bethe trees joined at the root</a>, Linear Algebra and its Appl., 428, 2008, 2961-2979.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F 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.
%e 7 is in the sequence because the corresponding rooted tree is Y, a generalized Bethe tree.
%p 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;
%t r[n_Integer] := r[n] = FactorInteger[n][[1, 1]];
%t s[n_Integer] := n/r[n];
%t 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];
%t A = {};
%t For[n = 1, n <= 3000, n++, If[Q[n] === 0, , Print[n, " ", Q[n]]; A = Union[A, {n}]]];
%t A (* _Jean-François Alcover_, Aug 03 2024, after _Emeric Deutsch_ *)
%Y Cf. A214578.
%Y Differs from A243497 for the first time at n=31.
%K nonn
%O 1,1
%A _Emeric Deutsch_, Aug 18 2012