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A184162 Number of chains in the rooted tree with Matula-Goebel number n. 1
1, 3, 7, 5, 15, 9, 11, 7, 13, 17, 31, 11, 19, 13, 21, 9, 23, 15, 15, 19, 17, 33, 27, 13, 29, 21, 19, 15, 35, 23, 63, 11, 37, 25, 25, 17, 23, 17, 25, 21, 39, 19, 27, 35, 27, 29, 43, 15, 21, 31, 29, 23, 19, 21, 45, 17, 21, 37, 47, 25, 31, 65, 23, 13, 33, 39, 31, 27, 33, 27, 39, 19, 35, 25, 35, 19, 41, 27, 67, 23 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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. A chain is a nonempty set of pairwise comparable vertices.

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.

REFERENCES

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 Review, 10, 1968, 273.

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

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

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

EXAMPLE

a(5) = 15 because the rooted tree with Matula-Goebel number 5 is a path ABCD on 4 vertices and any nonempty subset of {A,B,C,D} is a chain.

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+2*a(pi(n)) else a(r(n))+a(s(n))-1 end if end proc: seq(a(n), n = 1 .. 80);

PROG

(Haskell)

import Data.List (genericIndex)

a184162 n = genericIndex a184162_list (n - 1)

a184162_list = 1 : g 2 where

   g x = y : g (x + 1) where

     y = if t > 0 then 2 * a184162 t + 1 else a184162 r + a184162 s - 1

         where t = a049084 x; r = a020639 x; s = x `div` r

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS

Cf. A184160.

Cf. A049084, A020639.

Sequence in context: A268261 A090940 A090916 * A059912 A115765 A282598

Adjacent sequences:  A184159 A184160 A184161 * A184163 A184164 A184165

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 19 2011

STATUS

approved

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Last modified January 20 04:21 EST 2019. Contains 319323 sequences. (Running on oeis4.)