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A184170
Number of vertices that have largest escape distance in the rooted tree having Matula-Goebel number n.
1
1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 1, 1, 1, 2, 3, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 3, 1, 3, 3, 2, 1, 4, 1, 1, 1, 1, 1, 3, 1, 3, 2, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 3, 1, 2, 1, 1, 1, 3, 2, 2, 1, 4, 1, 1
OFFSET
1,6
COMMENTS
The escape distance of a vertex v in a rooted tree T is the distance from v to the nearest leaf of T that is a descendant of v. For the rooted tree ARBCDEF, rooted at R, the escape distance of B is 4 (the leaf A is not a descendant of B).
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
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 Review, 10, 1968, 273.
FORMULA
In A184167 one can find the generating polynomial P(n)=P(n,x) of the vertices of the rooted tree having Matula-Goebel number n, according to escape distance. a(n) is equal to the coefficient of the highest power of x.
EXAMPLE
a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 4 vertices with escape distances 0,0,1, and 2; the largest value (2) occurs only once.
a(15)=2 because the rooted tree with Matula number 15 is the path tree ABRCDE, rooted at R; the escape distances are 0,1,2,2,1,0; the largest value (2) occurs twice.
MAPLE
with(numtheory): P := proc (n) local r, s, LLL: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LLL := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+LLL(pi(n)) else min(LLL(r(n)), LLL(s(n))) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then P(pi(n))+x^(1+LLL(pi(n))) else P(r(n))+P(s(n))-x^max(LLL(r(n)), LLL(s(n))) end if end proc: a := proc (n) options operator, arrow: coeff(P(n), x, degree(P(n))) end proc: seq(a(n), n = 1 .. 110);
MATHEMATICA
r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
LLL[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, 1 + LLL[PrimePi[n]], True, Min[LLL[r[n]], LLL[s[n]]]];
P[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, P[PrimePi[n]] + x^(1 + LLL[PrimePi[n]]), True, P[r[n]] + P[s[n]] - x^Max[LLL[r[n]], LLL[s[n]]]];
a[n_] := Coefficient[P[n], x, Exponent[P[n], x]];
Table[a[n], {n, 1, 110}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)
CROSSREFS
Sequence in context: A326620 A353372 A305501 * A025919 A095684 A205565
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 23 2011
STATUS
approved