

A191515


Number of vertices of outdegree >=2 in the rooted tree having MatulaGoebel number n.


0



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
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OFFSET

1,14


COMMENTS

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.


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 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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Let g(n)=G(n,x) be the generating polynomial of the vertices of outdegree >=2 of the rooted tree having MatulaGoebel number n, with respect to level. Then g(1)=0; if n = p(t) (=the tth 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 MatulaGoebel number 5 is the pathtree on 4 vertices.
a(7)=1 because the rooted tree with MatulaGoebel 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



