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A184160 Number of root subtrees in the rooted tree with Matula-Goebel number n. 5
1, 2, 3, 4, 4, 6, 5, 8, 9, 8, 5, 12, 7, 10, 12, 16, 6, 18, 9, 16, 15, 10, 10, 24, 16, 14, 27, 20, 9, 24, 6, 32, 15, 12, 20, 36, 13, 18, 21, 32, 8, 30, 11, 20, 36, 20, 13, 48, 25, 32, 18, 28, 17, 54, 20, 40, 27, 18, 7, 48, 19, 12, 45, 64, 28, 30, 10, 24, 30, 40, 17, 72, 16, 26, 48, 36, 25, 42, 11, 64, 81, 16, 11, 60, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A root subtree of a rooted tree T is a subtree of T containing the root.

Also number of antichains in the rooted tree with Matula-Goebel number n. (The vertices of a rooted tree can be regarded as a partially ordered set, where u<=v holds for two vertices u and v if and only if u lies on the unique path between v and the root; an antichain is a nonempty set of mutually incomparable vertices). Example: a(5)=4 because the rooted tree with Matula-Goebel number 5 is a path ABCD and the only antichains are the 1-element subsets of {A,B,C,D}.

There is a simple bijection between the root subtrees and the antichains of a rooted tree: the leaves of a root subtree form an antichain (supplied by Stephan Wagner).

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

Alois P. Heinz, Table of n, a(n) for n = 1..10000

É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint 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. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.

Index entries for sequences related to Matula-Goebel numbers

FORMULA

a(1)=1; if n=p(t) (=the t-th prime), then a(n)=1+a(t); if n=rs (r,s,>=2), then a(n)=a(r)a(s). The Maple program is based on this recursive formula.

Completely multiplicative with a(prime(t)) = 1 + a(t). - Andrew Howroyd, Aug 01 2018

EXAMPLE

a(2^m) = 2^m because the rooted tree with Matula-Goebel number 2^m is a star with m edges (each edge can be included or not in the subtree).

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 1 elif bigomega(n) = 1 then 1+a(pi(n)) else a(r(n))*a(s(n)) end if end proc: seq(a(n), n = 1 .. 85);

MATHEMATICA

a[1] = 1; a[p_?PrimeQ] := a[p] = 1+a[PrimePi[p]]; a[n_] := a[n] = With[{f = FactorInteger[n]}, Times @@ ((a /@ f[[All, 1]])^f[[All, 2]])]; Array[a, 100] (* Jean-François Alcover, May 03 2017 *)

PROG

(PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); (1 + a(primepi(p)))^e)} \\ Andrew Howroyd, Aug 01 2018

CROSSREFS

Cf. A184161.

Sequence in context: A146288 A244361 A199424 * A064553 A126012 A283267

Adjacent sequences:  A184157 A184158 A184159 * A184161 A184162 A184163

KEYWORD

nonn,mult

AUTHOR

Emeric Deutsch, Oct 19 2011

STATUS

approved

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Last modified October 15 15:14 EDT 2019. Contains 328030 sequences. (Running on oeis4.)