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A184159 The difference between the levels of the highest and lowest leaves in the rooted tree with Matula-Goebel number n. 0
0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 1, 1, 1, 0, 0, 1, 0, 2, 0, 3, 0, 1, 0, 2, 0, 1, 2, 2, 0, 0, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 0, 2, 1, 2, 0, 1, 1, 1, 0, 3, 0, 2, 1, 4, 0, 0, 1, 3, 0, 2, 1, 2, 2, 1, 0, 2, 1, 1, 2, 2, 3, 2, 0, 3, 0, 1, 0, 2, 2, 3, 1, 2, 1, 2, 3, 3, 1, 1, 0, 1, 2, 2, 2, 2, 0, 2, 1, 1, 1, 1, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

COMMENTS

The Matula-Goebel number of a rooted tree is 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.

LINKS

Table of n, a(n) for n=1..110.

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011

FORMULA

In A184154 one constructs for each n the generating polynomial P(n,x) of the leaves of the rooted tree with Matula-Goebel number n, according to their levels. a(n) = degree of the numerator of P(n,1/x) (see the Maple program).

EXAMPLE

a(7)=0 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with all leaves at level 2.

a(2^m)=0 because the rooted tree with Matula-Goebel number 2^m is the star with m edges; all leaves are at level 1.

MAPLE

with(numtheory): a := proc (n) local r, s, P: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: P := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: degree(numer(subs(x = 1/x, P(n)))) end proc; seq(a(n), n = 1 .. 110);

CROSSREFS

Cf. A184154

Sequence in context: A141684 A152492 A075446 * A231122 A178686 A142724

Adjacent sequences:  A184156 A184157 A184158 * A184160 A184161 A184162

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 17 2011

STATUS

approved

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Last modified December 11 06:43 EST 2016. Contains 279041 sequences.