

A191517


Maximum edgedegree in the rooted tree with MatulaGoebel number n.


0



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
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OFFSET

2,4


COMMENTS

The degree of an edge is the number of edges adjacent to it.
The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


REFERENCES

A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355375, 1979. F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, WileyVCH, 2000.


LINKS

Table of n, a(n) for n=2..120.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288
Index entries for sequences related to MatulaGoebel numbers


FORMULA

In A191516 one finds the generating polynomial f(n)=F(n,x) of the edges of the rooted tree with MatulaGoebel number n, with respect to edgedegree. a(n)=degree of this polynomial.


EXAMPLE

a(7)=2 because the rooted tree with MatulaGoebel 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);


CROSSREFS

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



