A196062 - OEIS

A196062

Number of leaf-parents of the rooted tree with Matula-Goebel number n.

0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 3, 1, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 1, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 3, 1, 4, 2, 2, 2, 3, 1, 3, 3, 2, 3, 1, 3, 3, 1, 2, 3, 3, 2, 3, 2, 3, 3, 2, 2, 4, 2, 2, 4, 3, 2, 3, 2, 3, 3, 2, 2, 4, 3, 3, 2, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 4, 2, 3

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OFFSET

1,6

COMMENTS

A leaf-parent in a rooted tree is a node that is the parent of at least one leaf.

The Matula-Goebel number of a rooted tree is 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

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

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

FORMULA

a(1)=0; a(2)=1; if n=prime(t) (the t-th prime, t>1), then a(n)=a(t); if n=rs (r,s >=2) and both r and s are even, then a(n)=a(r)+a(s)-1; if n=r*s (r,s >=2) and not both r and s are even, then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.

EXAMPLE

a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.

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 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n)) elif `mod`(r(n), 2) = 0 and `mod`(s(n), 2) = 0 then a(r(n))+a(s(n))-1 else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);

MATHEMATICA

r[n_] := FactorInteger[n][[1, 1]];

s[n_] := n/r[n];

a[n_] := Which[n == 1, 0, n == 2, 1, PrimeOmega[n] == 1, a[PrimePi[n]], Mod[r[n], 2] == 0 && Mod[s[n], 2] == 0, a[r[n]] + a[s[n]] - 1, True, a[r[n]] + a[s[n]]];

Table[a[n], {n, 1, 110}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)

PROG

(Haskell)

import Data.List (genericIndex)

a196062 n = genericIndex a196062_list (n - 1)

a196062_list = 0 : 1 : g 3 where

g x = y : g (x + 1) where

y | t > 0 = a196062 t

| otherwise = a196062 r + a196062 s - 0 ^ (x `mod` 4)

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

-- Reinhard Zumkeller, Sep 03 2013

CROSSREFS