A184160 Number of root subtrees in the rooted tree with Matula-Goebel number n.

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

Offset: 1

Author

Emeric Deutsch, Oct 19 2011

Keywords

• nonn
• mult

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.

Examples

			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).
		

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.
• Emeric 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

Crossrefs

Cf. A184161.

Programs

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 *)

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

Formula

a(1)=1; if n=prime(t), then a(n)=1+a(t); if n=r*s (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