A191515 Number of vertices of outdegree >=2 in the rooted tree having Matula-Goebel number n.

0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 1, 1, 3, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 1, 1, 2, 1, 2, 2, 3

Offset

1,14

Comments

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.

Links

Table of n, a(n) for n=1..86.

Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.

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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.

Index entries for sequences related to Matula-Goebel numbers

Formula

Let g(n)=G(n,x) be the generating polynomial of the vertices of outdegree >=2 of the rooted tree having Matula-Goebel number n, with respect to level. Then g(1)=0; if n = p(t) (=the t-th prime), then g(n)=x*g(t); if n=rs (r,s>=2), then g(n)=1+g(r)+g(s)-G(r,0)-G(s,0). Clearly, a(n)=G(n,1).

Example

a(5)=0 because the rooted tree with Matula-Goebel number 5 is the path-tree on 4 vertices.
a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.

Prog

(PARI) a(n) = my(f=factor(n)); (vecsum(f[, 2])>=2) + [self()(primepi(p))|p<-f[, 1]]*f[, 2]; \\ Kevin Ryde, Oct 30 2021

Crossrefs

Cf. A007097 (indices of 0's).

Sequence in context: A294623 A337473 A039738 * A320001 A168201 A263000

Adjacent sequences: A191512 A191513 A191514 * A191516 A191517 A191518

Keyword

nonn

Author

Emeric Deutsch, Dec 10 2011

Status

approved