A191517 Maximum edge-degree in the rooted tree with Matula-Goebel number n.

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, 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, 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

Offset

2,4

Comments

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

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.

References

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

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

Links

Table of n, a(n) for n=2..120.

Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.

F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

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.

Index entries for sequences related to Matula-Goebel numbers

Formula

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.

Example

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

Maple

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);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
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]]];
h[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*g[PrimePi[n]] + h[PrimePi[n]], True, h[r[n]] + h[s[n]]];
f[n_] := g[n] + h[n];
a[n_] := Exponent[f[n], x];
Table[a[n], {n, 2, 120}] (* Jean-François Alcover, Jun 20 2024, after Maple code *)

Crossrefs

Cf. A191516.

Sequence in context: A177227 A174373 A232270 * A303370 A082528 A055980

Adjacent sequences: A191514 A191515 A191516 * A191518 A191519 A191520

Keyword

nonn

Author

Emeric Deutsch, Dec 15 2011

Status

approved