A184162 Number of chains in the rooted tree with Matula-Goebel number n.

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

Offset: 1

Author

Emeric Deutsch, Oct 19 2011

Keywords

• nonn

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.

Examples

			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.
		

Links

• Reinhard Zumkeller, 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.
• 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. A109082 (height), A196056 (vertices at levels).

Cf. A184160 (antichains).

Cf. A049084, A020639.

Programs

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

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);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
a[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + 2*a[PrimePi[n]], True, a[r[n]] + a[s[n]] - 1];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 24 2024, after Maple code *)

PARI

a(n) = my(f=factor(n)); [self()(primepi(p)) |p<-f[,1]] * f[,2]*2 + 1; \\ Kevin Ryde, Aug 25 2021

Formula

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

a(n) = 1 + Sum_{k=1..A109082(n)} A196056(n,k)*2^k. - Kevin Ryde, Aug 25 2021