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A196067 Number of pendant vertices in the rooted tree with Matula-Goebel number n. 0
0, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 2, 4, 3, 3, 4, 3, 3, 2, 3, 4, 2, 3, 3, 4, 3, 3, 2, 5, 2, 3, 3, 4, 4, 4, 3, 4, 3, 4, 4, 3, 3, 3, 3, 5, 4, 3, 3, 4, 5, 4, 2, 5, 4, 3, 3, 4, 4, 2, 4, 6, 3, 3, 4, 4, 3, 4, 4, 5, 4, 4, 3, 5, 3, 4, 3, 5, 4, 3, 3, 5, 3, 4, 3, 4, 5, 4, 4, 4, 2, 3, 4, 6, 3, 5, 3, 4, 4, 4, 4, 5, 4, 5, 5, 5, 3, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A pendant vertex in a tree is a vertex having degree 1.

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.

LINKS

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

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

Index entries for sequences related to Matula-Goebel numbers

FORMULA

a(1)=0; a(2)=2; if n=p(t) (=the t-th prime) and t is prime, then a(n)=a(t); if n=p(t) (the t-th prime) and t is not prime, then a(n)=1+a(t); if n=rs (r,s,>=2), then a(n)=a(r)+a(s)-the number of primes in {r,s}. The Maple program is based on this recursive formula.

EXAMPLE

a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.

a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

MAPLE

with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif n = 2 then 2 elif bigomega(n) = 1 and bigomega(pi(n)) = 1 then a(pi(n)) elif bigomega(n) = 1 then a(pi(n))+1 elif bigomega(r(n)) = 1 and bigomega(s(n)) = 1 then a(r(n))+a(s(n))-2 elif min(bigomega(r(n)), bigomega(s(n))) = 1 then a(r(n))+a(s(n))-1 else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);

CROSSREFS

Sequence in context: A067743 A029230 A280945 * A251141 A319696 A320111

Adjacent sequences:  A196064 A196065 A196066 * A196068 A196069 A196070

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 03 2011

STATUS

approved

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Last modified December 16 12:37 EST 2018. Contains 318160 sequences. (Running on oeis4.)