A196048 -id:A196048 - cp's OEIS frontend

A196047 Path length of the rooted tree with Matula-Goebel number n.

0, 1, 3, 2, 6, 4, 5, 3, 6, 7, 10, 5, 8, 6, 9, 4, 9, 7, 7, 8, 8, 11, 11, 6, 12, 9, 9, 7, 12, 10, 15, 5, 13, 10, 11, 8, 10, 8, 11, 9, 13, 9, 11, 12, 12, 12, 15, 7, 10, 13, 12, 10, 9, 10, 16, 8, 10, 13, 14, 11, 13, 16, 11, 6, 14, 14, 12, 11, 14, 12, 14, 9, 14, 11, 15, 9, 15, 12, 17, 10, 12, 14, 17, 10, 15, 12, 15, 13, 12, 13, 13, 13, 18, 16, 13, 8, 19, 11, 16, 14

Offset: 1

Emeric Deutsch, Sep 27 2011

nonn

The path length of a rooted tree is defined as the sum of distances of all nodes to the root of the tree.

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.

			a(7) = 5 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (0+1+2+2 = 5).
a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A049084, A020639.

Cf. A196048, A343006.

import Data.List (genericIndex)
a196047 n = genericIndex a196047_list (n - 1)
a196047_list = 0 : g 2 where
   g x = y : g (x + 1) where
     y = if t > 0 then a196047 t + a061775 t else a196047 r + a196047 s
         where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := proc (n) local r, s, N: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: N := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+N(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 100);

a[m_] := Module[{r, s, Nn},
   r[n_] := FactorInteger[n][[1, 1]];
   s[n_] := n/r[n];
   Nn[n_] := Which[n == 1, 1,
      PrimeOmega[n] == 1, 1+Nn[PrimePi[n]],
      True, Nn[r[n]]+Nn[s[n]]-1];
   Which[m == 1, 0,
   PrimeOmega[m] == 1, a[PrimePi[m]]+Nn[PrimePi[m]],
   True, a[r[m]]+a[s[m]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 03 2023, after Maple code *)

NPl(n) = { if(n==1, return([1,0]),
    my(f=factor(n)~, v=Mat(vector(#f,k,NPl(primepi(f[1,k]))~))  );
    return( [ 1+sum(k=1,#f,v[1,k]*f[2,k]) , sum(k=1,#f,(v[1,k]+v[2,k])*f[2,k]) ] ) )
  };
A196047(n) = NPl(n)[2]; \\ François Marques, Apr 02 2021

a(1)=0; if n=prime(t) then a(n)=a(t)+N(t), where N(t) is the number of nodes of the rooted tree with Matula number t; if n=r*s (r,s>=2), then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.

a(n) = A196048(n) + A343006(n). - François Marques, Apr 02 2021

A348959 Childless terminal Wiener index of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 2, 0, 3, 2, 6, 4, 4, 0, 8, 3, 8, 5, 12, 2, 10, 6, 10, 10, 5, 4, 15, 6, 10, 12, 16, 4, 12, 0, 20, 6, 10, 12, 18, 8, 15, 12, 18, 3, 19, 8, 12, 14, 12, 5, 24, 20, 14, 12, 19, 12, 21, 7, 26, 18, 12, 2, 21, 10, 6, 22, 30, 14, 14, 6, 20, 14, 22, 10, 28, 10

Offset: 1

Kevin Ryde, Nov 05 2021

nonn

This is a variation on the terminal Wiener index defined by Gutman, Furtula, and Petrović. Here terminal vertices are taken as the childless vertices, so a(n) is the sum of the path lengths between pairs of childless vertices.

This sequence differs from the free tree form A196055 when n is prime, since n prime means the root is degree 1 so is a terminal vertex for A196055 but not here.

Kevin Ryde, Table of n, a(n) for n = 1..10000
F. Goebel, On a 1-1-Correspondence between Rooted Trees and Natural Numbers, Journal of Combinatorial Theory, series B, volume 29, 1980, pages 141-143.
Ivan Gutman, Boris Furtula and Miroslav Petrović, Terminal Wiener Index, Journal of Mathematical Chemistry, volume 46, 2009, pages 522-531.
D. W. Matula, A Natural Rooted Tree Enumeration By Prime Factorization, SIAM Review, volume 10, number 2, April 1968, page 273 (also at JSTOR).
Kevin Ryde, PARI/GP Code and Notes
Index entries for sequences related to Matula-Goebel numbers

Cf. A196055 (free tree), A196048 (external path length), A109129 (childless vertices), A288469 (unplant).

Cf. A027746 (prime factorization).

PARI
```
\\ See links.
```

a(n) = Sum_{j=1..k} (a(primepi(p[j])) + E(p[j])*(C(n)-C(p[j]))), where n = p[1]*...*p[k] is the prime factorization of n with multiplicity (A027746), E(n) = A196048(n) is external path length, and C(n) = A109129(n) is number of childless vertices.
a(n) = A196055(n) - (A196048(n) if n prime).
a(n) = A196055(A288469(n)).

A196055 The terminal Wiener index of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 6, 6, 4, 4, 4, 8, 8, 8, 5, 12, 8, 10, 12, 10, 10, 5, 10, 15, 6, 10, 12, 16, 10, 12, 5, 20, 6, 10, 12, 18, 15, 15, 12, 18, 10, 19, 16, 12, 14, 12, 12, 24, 20, 14, 12, 19, 20, 21, 7, 26, 18, 12, 10, 21, 18, 6, 22, 30, 14, 14, 15, 20, 14, 22, 18, 28, 19, 18, 16, 26, 14, 22, 12, 28, 24, 12, 12, 30, 14, 19, 14, 21, 24, 24

Offset: 1

Emeric Deutsch, Sep 29 2011

nonn

The terminal Wiener index of a connected graph is the sum of the distances between all pairs of nodes of degree 1.

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.

			a(7)=6 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (2+2+2=6).

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, B. Furtula and M. Petrovic, Terminal Wiener index, J. Math. Chem., 46. 2009, 522-531.
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

Cf. A109129, A196048, A196051, A348959.

with(numtheory): TW := proc (n) local r, s, LV, EPL, Tw: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: LV := proc (n) if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then LV(pi(n)) else LV(r(n))+LV(s(n)) end if end proc: EPL := proc (n) if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then EPL(pi(n))+LV(pi(n)) else EPL(r(n))+EPL(s(n)) end if end proc: Tw := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then Tw(pi(n)) else Tw(r(n))+Tw(s(n))+EPL(r(n))*LV(s(n))+EPL(s(n))*LV(r(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = 1 then TW(pi(n))+LV(pi(n)) elif bigomega(n) = 1 then TW(pi(n))+EPL(n) else Tw(r(n))+Tw(s(n))+EPL(r(n))*LV(s(n))+EPL(s(n))*LV(r(n)) end if end proc; seq(TW(n), n = 1 .. 90);

Let LV(m) and EPL(m) denote the number of leaves and the external path length, respectively, of the rooted tree with Matula number m (see A109129 and A196048, where LV(m) and EPL(m) are obtained recursively). a(1)=0; if n=p(t) (=the t-th prime) and t is prime, then a(n) = a(t) + LV(t); if n=p(t) (=the t-th prime) and t is not prime, then a(n) = a(t) + LV(t) + EPL(t). Now assume that n is not prime; it can be written n=rs, where r is prime and s >= 2. If s is prime, then a(n) = a(r) - EPL(r) + a(s) - EPL(s) + EPL(r)*LV(s) + EPL(s)*LV(r); if s is not prime, then a(n) = a(r) - EPL(r) + a(s) + EPL(r)*LV(s) + EPL(s)*LV(r); the Maple program is based on this recursive formula.

If m > 2 then a(2^m) = m(m-1) because the rooted tree with Matula-Goebel number 2^m is a star with m edges and the vertices of each of the binomial(m,2) pairs of nodes of degree 1 are at distance 2.

A343006 Internal path length of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 1, 0, 3, 1, 1, 0, 2, 3, 6, 1, 3, 1, 4, 0, 3, 2, 1, 3, 2, 6, 5, 1, 6, 3, 3, 1, 6, 4, 10, 0, 7, 3, 4, 2, 3, 1, 4, 3, 6, 2, 3, 6, 5, 5, 8, 1, 2, 6, 4, 3, 1, 3, 9, 1, 2, 6, 6, 4, 5, 10, 3, 0, 6, 7, 3, 3, 6, 4, 6, 2, 5, 3, 7, 1, 7, 4, 10, 3, 4, 6, 9, 2, 6, 3, 7, 6, 3, 5, 4, 5, 11, 8, 4, 1, 11, 2, 8, 6

Offset: 1

François Marques, Apr 02 2021

nonn

The internal path length of a rooted tree is defined as the sum of the distances of all internal nodes to the root.

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.

			a(7) = 1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (0+1).
a(2^m) = 0 because the rooted tree with Matula-Goebel number 2^m is a star with m edges and therefore has only one internal node: its root.
a(3^m) = m because the rooted tree with Matula-Goebel number 3^m is a star with m branches of length 2, so the internal nodes are the root and the m nodes attached to it.

E. 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.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A196047, A196048.

InIpl(n)={ if(n==1, return([0,0]),
    my(f=factor(n)~, v=Mat(vector(#f,k,InIpl(primepi(f[1,k]))~))  );
    return( [ 1+sum(k=1,#f,v[1,k]*f[2,k]) , sum(k=1,#f,(v[1,k]+v[2,k])*f[2,k]) ] ) )
};
A343006(n) = InIpl(n)[2];

a(n) = A196047(n) - A196048(n).

a(r*s) = a(r) + a(s).

A352288 Total cophenetic index of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 2, 0, 3, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 2, 1, 0, 3, 3, 1, 0, 2, 1, 4, 0, 0, 1, 1, 0, 2, 0, 2, 1, 6, 0, 0, 1, 3, 1, 3, 0, 3, 0, 1, 0, 1, 0, 6, 2, 1, 1, 3, 0, 4, 3, 0, 3, 1, 1, 1, 0, 0, 2, 2, 1, 2, 4, 1

Offset: 1

Kevin Ryde, Mar 16 2022

nonn

Mir, Rosselló, and Rotger, define the cophenetic value of a pair of childless vertices as the depth (distance down from the root) of their deepest common ancestor, and they then define the total cophenetic index of a tree as the sum of the cophenetic values over all such pairs.
a(n) = 0 iff n is in A325663, being rooted stars with any arm lengths, since the root (depth 0) is the common ancestor of every childless pair.
An identity relating the childless terminal Wiener index TW(n) = A348959(n) can be constructed by noting it measures distances from a pair of childless vertices to their common ancestor, and the cophenetic values measure from that ancestor up to the root.  So 2*a(n) + TW(n) is total depths Ext(n) = A196048(n) of the childless vertices, repeated by childless vertices C(n) = A109129(n) except itself, so that 2*a(n) + TW(n) = Ext(n)*(C(n) - 1)

			For n=111, the tree and its childless pairs and deepest common ancestors are
  root  R         pair  ancestor depth
       / \         G,D     A       1
      A   B        G,E     A       1
     /|\   \       D,E     A       1
    C D E   F     any,F    R       0
    |                             ---
    G                 total a(n) = 3

Arnau Mir, Francesc Rosselló, and Lucía Rotger, A New Balance Index for Phylogenetic Trees, arXiv:1202.1223 [q-bio.PE], 2012.
Kevin Ryde, PARI/GP Code
Index entries for sequences related to Matula-Goebel numbers

Cf. A348959 (terminal Wiener), A196048 (external length), A109129 (childless vertices).

Cf. A325663 (indices of 0's), A352289 (max by leaves).

PARI
```
\\ See links.
```

a(n) = Sum_{i=1..k} a(primepi(p[i])) + binomial(C(p[i]),2), where n = p[1]*...*p[k] is the prime factorization of n with multiplicity (A027746), and C(n) = A109129(n) is the number of childless vertices.

cp's OEIS Frontend

A196047 Path length of the rooted tree with Matula-Goebel number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A348959 Childless terminal Wiener index of the rooted tree with Matula-Goebel number n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A196055 The terminal Wiener index of the rooted tree with Matula-Goebel number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Formula

A343006 Internal path length of the rooted tree with Matula-Goebel number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A352288 Total cophenetic index of the rooted tree with Matula-Goebel number n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula