A214567 Maximal number of distinct rooted trees obtained from the rooted tree with Matula-Goebel number n by adding one pendant edge at one of its vertices.

1, 2, 3, 2, 4, 4, 3, 2, 3, 5, 5, 4, 5, 4, 6, 2, 4, 4, 3, 5, 5, 6, 4, 4, 4, 6, 3, 4, 6, 7, 6, 2, 7, 5, 6, 4, 5, 4, 7, 5, 6, 6, 5, 6, 6, 5, 7, 4, 3, 5, 6, 6, 3, 4, 8, 4, 5, 7, 5, 7, 5, 7, 5, 2, 8, 8, 4, 5, 6, 7, 6, 4, 6, 6, 6, 4, 7, 8, 7, 5, 3

Offset: 1

Author

Emeric Deutsch, Jul 25 2012

Keywords

• nonn

Comments

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.

Sum_{j such that rooted tree with Matula-Goebel number j has n vertices} a(j) = A000107(n). Example: the Matula-Goebel numbers of the rooted trees with 4 vertices are 5,6,7,8 and a(5)+a(6)+a(7)+a(8) = 4+4+3+2=13 = A000107(4).

Examples

			a(4)=2 because the rooted tree with Matula-Goebel number 4 is V; adding an edge at either of the two leaves yields the same rooted tree.
a(5)=4 because the rooted tree with Matula-Goebel number 5 is the path on 4 vertices; adding one edge at any of the vertices yields a new rooted tree.
a(987654321)=18 (reader may verify this on Fig. 2 of the Deutsch paper).
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Emeric Deutsch, Rooted tree statistics from Matula numbers, 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. A000107.

Cf. A049084, A027748.

Programs

Haskell

import Data.List (genericIndex)
a214567 n = genericIndex a214567_list (n - 1)
a214567_list = 1 : g 2 where
  g x = y : g (x + 1) where
    y | t > 0     = a214567 t + 1
      | otherwise = 1 + sum (map ((subtract 1) . a214567) $ a027748_row x)
       where t = a049084 x
-- Reinhard Zumkeller, Sep 03 2013

Maple

with(numtheory): a := proc (n) local FS: FS := proc (n) options operator, arrow: factorset(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+a(pi(n)) else 1+add(a(FS(n)[j])-1, j = 1 .. nops(FS(n))) end if end proc: seq(a(n), n = 1 .. 130);

Mathematica

a[n_] := Which[n == 1, 1, PrimeQ[n], 1 + a[PrimePi[n]], True, 1 + Total[a[#] - 1& /@ FactorInteger[n][[All, 1]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 25 2024 *)

PARI

a(n) = 1 + vecsum([self()(primepi(p)) |p<-factor(n)[,1]]); \\ Kevin Ryde, Oct 19 2022

Formula

a(1)=1; if n is t-th prime, then a(n)=1+a(t); if n is composite, then a(n) = 1+Sum_{p|n}(a(p)-1), where summation is over the distinct prime divisors of n.