A191517 - OEIS

%I #22 Jun 21 2024 07:34:30

%S 0,1,1,2,2,2,2,2,2,2,3,3,3,2,3,3,3,3,3,3,2,3,4,2,3,3,4,3,3,2,4,2,3,3,

%T 4,4,4,3,4,3,4,4,3,3,3,3,5,3,3,3,4,4,4,2,5,4,3,3,4,4,2,4,5,3,3,4,3,3,

%U 4,4,5,4,4,3,5,3,4,3,5,4,3,3,5,3,4,3,4,5,4,3,4,2,3,4,6,3,4,3,4,4,3,4,5,4,5,5,5,3,3,4,6,4,5,3,4,4,3,3,5

%N Maximum edge-degree in the rooted tree with Matula-Goebel number n.

%C The degree of an edge is the number of edges adjacent to it.

%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.

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

%D R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.

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

%F In A191516 one finds the generating polynomial f(n)=F(n,x) of the edges of the rooted tree with Matula-Goebel number n, with respect to edge-degree. a(n)=degree of this polynomial.

%e a(7)=2 because the rooted tree with Matula-Goebel number 7 is Y; all edges have degree 2.

%p with(numtheory): f := proc (n) local r, s, g, h: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: g := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then x^bigomega(pi(n)) else x^bigomega(s(n))*g(r(n))+x^bigomega(r(n))*g(s(n)) end if end proc: h := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then x*g(pi(n))+h(pi(n)) else h(r(n))+h(s(n)) end if end proc: sort(expand(g(n)+h(n))) end proc: seq(degree(f(n)), n = 2 .. 120);

%t r[n_] := FactorInteger[n][[1, 1]];

%t s[n_] := n/r[n];

%t g[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x^(PrimeOmega[PrimePi[n]]), True, x^(PrimeOmega[s[n]])*g[r[n]] + x^(PrimeOmega[r[n]])*g[s[n]]];

%t h[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*g[PrimePi[n]] + h[PrimePi[n]], True, h[r[n]] + h[s[n]]];

%t f[n_] := g[n] + h[n];

%t a[n_] := Exponent[f[n], x];

%t Table[a[n], {n, 2, 120}] (* _Jean-François Alcover_, Jun 20 2024, after Maple code *)

%Y Cf. A191516.

%K nonn

%O 2,4

%A _Emeric Deutsch_, Dec 15 2011