

A184161


Number of subtrees in the rooted tree with MatulaGoebel number n.


4



1, 3, 6, 6, 10, 10, 11, 11, 15, 15, 15, 17, 17, 17, 21, 20, 17, 25, 20, 24, 24, 21, 25, 30, 28, 25, 36, 28, 24, 34, 21, 37, 28, 24, 32, 44, 30, 30, 34, 41, 25, 40, 28, 32, 48, 36, 34, 55, 37, 45, 32, 40, 37, 64, 36, 49, 41, 34, 24, 59, 44, 28, 57, 70, 44, 44, 30, 37, 48, 53, 41, 81, 40, 44, 63, 49, 41, 56, 32, 74
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OFFSET

1,2


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..80.
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 33143319.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint 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. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Let b(n)=A184160(n) denote the number of those subtrees of the rooted tree with MatulaGoebel number n that contain the root. Then a(1)=1; if n=p(t) (=the t=th prime), then a(n)=1+a(t)+b(t); if n=rs (r,s,>=2), then a(n)=a(r)+a(s)+(b(r)1)(b(s)1)1. The Maple program is based on this recursive formula.


EXAMPLE

a(4) = 6 because the rooted tree with MatulaGoebel number 4 is V; it has 6 subtrees (three 1vertex subtrees, two 1edge subtrees, and the tree itself).


MAPLE

with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+b(pi(n)) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n))+b(pi(n))+1 else a(r(n))+a(s(n))+(b(r(n))1)*(b(s(n))1)1 end if end proc: seq(a(n), n = 1 .. 80);


CROSSREFS

Cf. A184160 (subtrees containing the root), A184164 (numbers not occurring as terms).
Sequence in context: A108850 A111652 A159787 * A276000 A333616 A316563
Adjacent sequences: A184158 A184159 A184160 * A184162 A184163 A184164


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 19 2011


STATUS

approved



