

A196067


Number of pendant vertices in the rooted tree with MatulaGoebel 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
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OFFSET

1,2


COMMENTS

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

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.


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 MatulaGoebel numbers


FORMULA

a(1)=0; a(2)=2; if n=p(t) (=the tth prime) and t is prime, then a(n)=a(t); if n=p(t) (the tth 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 MatulaGoebel number 7 is the rooted tree Y.
a(2^m) = m because the rooted tree with MatulaGoebel 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



