A196047 -id:A196047 - cp's OEIS frontend

A061775 Number of nodes in rooted tree with Matula-Goebel number n.

1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 6, 5, 5, 6, 5, 6, 6, 6, 6, 6, 7, 6, 7, 6, 6, 7, 6, 6, 7, 6, 7, 7, 6, 6, 7, 7, 6, 7, 6, 7, 8, 7, 7, 7, 7, 8, 7, 7, 6, 8, 8, 7, 7, 7, 6, 8, 7, 7, 8, 7, 8, 8, 6, 7, 8, 8, 7, 8, 7, 7, 9, 7, 8, 8, 7, 8, 9, 7, 7, 8, 8, 7, 8, 8, 7, 9, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 7, 8, 8, 8, 9, 7, 7, 9

Offset: 1

N. J. A. Sloane, Jun 22 2001

nonn

Let p(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product p(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration).

Each n occurs A000081(n) times.

			a(4) = 3 because the rooted tree corresponding to the Matula-Goebel number 4 is "V", which has one root-node and two leaf-nodes, three in total.
See also the illustrations in A061773.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Keith Briggs, Matula numbers and rooted trees
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

One more than A196050.
Sum of entries in row n of irregular table A214573.
Number of entries in row n of irregular tables A182907, A206491, A206495 and A212620.
One less than the number of entries in row n of irregular tables A184187, A193401 and A193403.
Cf. A005517 (the position of the first occurrence of n).
Cf. A005518 (the position of the last occurrence of n).
Cf. A091233 (their difference plus one).
Cf. A214572 (Numbers k such that a(k) = 8).
Cf. also A000081, A061773, A049084, A020639, A049076, A078442, A091238, A091204, A091205, A109082, A127301, A109129, A193402, A193405, A193406, A196047, A196068, A198333, A206487, A206494, A206496, A214569, A214571, A213670, A214568, A228599, A245817, A245818.
Cf. also A075166, A075167, A216648, A216649.

import Data.List (genericIndex)
a061775 n = genericIndex a061775_list (n - 1)
a061775_list = 1 : g 2 where
   g x = y : g (x + 1) where
      y = if t > 0 then a061775 t + 1 else a061775 u + a061775 v - 1
          where t = a049084 x; u = a020639 x; v = x `div` u
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := proc (n) local u, v: u := n-> op(1, factorset(n)): v := n-> n/u(n): if n = 1 then 1 elif isprime(n) then 1+a(pi(n)) else a(u(n))+a(v(n))-1 end if end proc: seq(a(n), n = 1..108); # Emeric Deutsch, Sep 19 2011

a[n_] := Module[{u, v}, u = FactorInteger[#][[1, 1]]&; v = #/u[#]&; If[n == 1, 1, If[PrimeQ[n], 1+a[PrimePi[n]], a[u[n]]+a[v[n]]-1]]]; Table[a[n], {n, 108}] (* Jean-François Alcover, Jan 16 2014, after Emeric Deutsch *)

A061775(n) = if(1==n, 1, if(isprime(n), 1+A061775(primepi(n)), {my(pfs,t,i); pfs=factor(n); pfs[,1]=apply(t->A061775(t),pfs[,1]); (1-bigomega(n)) + sum(i=1, omega(n), pfs[i,1]*pfs[i,2])}));
for(n=1, 10000, write("b061775.txt", n, " ", A061775(n)));
\\ Antti Karttunen, Aug 16 2014

from functools import lru_cache
from sympy import isprime, factorint, primepi
@lru_cache(maxsize=None)
def A061775(n):
    if n == 1: return 1
    if isprime(n): return 1+A061775(primepi(n))
    return 1+sum(e*(A061775(p)-1) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022

a(1) = 1; if n = p_t (= the t-th prime), then a(n) = 1+a(t); if n = uv (u,v>=2), then a(n) = a(u)+a(v)-1.
a(n) = A091238(A091204(n)). - Antti Karttunen, Jan 2004
a(n) = A196050(n)+1. - Antti Karttunen, Aug 16 2014

More terms from David W. Wilson, Jun 25 2001

Extended by Emeric Deutsch, Sep 19 2011

A196048 External path length of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

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

Offset: 1

Emeric Deutsch, Sep 27 2011

nonn

The external path length of a rooted tree is defined as the sum of the distances of all leaves 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)=4 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (2+2=4).

François Marques, 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. A109129, A196047.

Cf. A049084, A020639.

import Data.List (genericIndex)
a196048 n = genericIndex a196048_list (n - 1)
a196048_list = 0 : 1 : g 3 where
   g x = y : g (x + 1) where
     y = if t > 0 then a196048 t + a109129 t else a196048 r + a196048 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, LV: 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: if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n))+LV(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);

a[m_] := Module[{r, s, LV},
   r[n_] := FactorInteger[n][[1, 1]];
   s[n_] := n/r[n];
   LV [n_] := Which[
      n == 1, 0,
      n == 2, 1,
      PrimeOmega[n] == 1, LV[PrimePi[n]],
      True, LV[r[n]] + LV[s[n]]];
   Which[
      m == 1, 0,
      m == 2, 1,
      PrimeOmega[m] == 1, a[PrimePi[m]] + LV[PrimePi[m]],
      True, a[r[m]] + a[s[m]]]];
Table[a[n], {n, 1, 110}] (* Jean-François Alcover, May 04 2023, after Maple code *)

LEpl(n) = { if(n==1, return([1,0]),
    my(f=factor(n)~, l, e, le);
      foreach(f,p,
        le=LEpl(primepi(p[1]));
        l+=le[1]*p[2];
        e+=(le[1]+le[2])*p[2];
      );
    return([l,e]) )
  };
A196048(n) = LEpl(n)[2]; \\ François Marques, Mar 14 2021

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

a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

Offset fixed by Reinhard Zumkeller, Sep 03 2013

A196051 The Wiener index of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 1, 4, 4, 10, 10, 9, 9, 20, 20, 20, 18, 18, 18, 35, 16, 18, 31, 16, 32, 32, 35, 31, 28, 56, 31, 48, 29, 32, 50, 35, 25, 56, 32, 52, 44, 28, 28, 50, 46, 31, 46, 29, 52, 72, 48, 50, 40, 48, 75, 52, 46, 25, 64, 84, 42, 46, 50, 32, 67, 44, 56, 67, 36, 76, 76, 28, 48, 72, 70, 46, 59, 46, 44, 102, 42, 79, 68, 52, 62, 88, 50, 48, 62, 79, 46, 75, 71, 40, 92, 71, 67, 84, 72, 71, 54, 75, 65, 104, 96

Offset: 1

Emeric Deutsch, Sep 27 2011

nonn

The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.

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)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+1+1+2+2+2=9).
a(2^m) = m^2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges and we have m distances 1 and m(m-1)/2 distances 2; m + m(m-1)=m^2.

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

Cf. A196047, A196050.
Cf. A049084, A020639.
Terminal Wiener indices: A196055, A348959.

import Data.List (genericIndex)
a196051 n = genericIndex a196051_list (n - 1)
a196051_list = 0 : g 2 where
   g x = y : g (x + 1) where
     y | t > 0     = a196051 t + a196047 t + a196050 t + 1
       | otherwise = a196051 r + a196051 s +
                     a196047 r * a196050 s + a196047 s * a196050 r
       where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

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

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
e[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, 1 + e[PrimePi[n]], True, e[r[n]] + e[s[n]]];
PL[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, 1 + e[PrimePi[n]] + PL[PrimePi[n]], True, PL[r[n]] + PL[s[n]]];
a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, a[PrimePi[n]] + PL[PrimePi[n]] + 1 + e[PrimePi[n]], True, a[r[n]] + a[s[n]] + PL[r[n]]*e[s[n]] + PL[s[n]]*e[r[n]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)

a(1)=0; if n = prime(t) (the t-th prime), then a(n)=a(t)+PL(t)+E(t)+1; if n=rs (r,s>=2), then a(n)=a(r)+a(s)+PL(r)E(s)+PL(s)E(r); PL(m) and E(m) denote the path length and the number of edges of the rooted tree with Matula number m (see A196047, A196050). The Maple program is based on this recursive formula.

A196068 Visitation length of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 3, 6, 5, 10, 8, 9, 7, 11, 12, 15, 10, 13, 11, 15, 9, 14, 13, 12, 14, 14, 17, 17, 12, 19, 15, 16, 13, 18, 17, 21, 11, 20, 16, 18, 15, 16, 14, 18, 16, 19, 16, 17, 19, 20, 19, 22, 14, 17, 21, 19, 17, 15, 18, 24, 15, 17, 20, 20, 19, 20, 23, 19, 13, 22, 22, 18, 18, 22, 20, 21, 17, 21, 18, 24, 16, 23, 20, 24, 18

Offset: 1

Emeric Deutsch, Oct 04 2011

nonn

The visitation length of a rooted tree is defined as the sum of the path length and the number of vertices. The path length of a rooted tree is defined as the sum of distances of all vertices to the root of the tree (see the Keijzer et al. reference).

			a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+2+2+4=9).
a(2^m) = 2m+1 because the rooted tree with Matula-Goebel number 2^m is a star with m edges (m+(m+1)=2m+1).

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.
M. Keijzer and J. Foster, Crossover bias in genetic programming, Lecture Notes in Computer Sciences, 4445, 2007, 33-44.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

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.
Index entries for sequences related to Matula-Goebel numbers

Cf. A196046, A196047.

Cf. A049084, A020639, A061775.

import Data.List (genericIndex)
a196068 n = genericIndex a196068_list (n - 1)
a196068_list = 1 : g 2 where
   g x = y : g (x + 1) where
     y | t > 0     = a196068 t + a061775 t + 1
       | otherwise = a196068 r + a196068 s - 1
       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 1 elif bigomega(n) = 1 then a(pi(n))+N(pi(n))+1 else a(r(n))+a(s(n))-1 end if end proc: seq(a(n), n = 1 .. 80);

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];
a[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, a[PrimePi[n]] + nn[PrimePi[n]] + 1, True, a[r[n]] + a[s[n]] - 1];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 25 2024, after Maple code *)

a(1)=1; if n=prime(t) (= the t-th prime) then a(n)=a(t)+N(t)+1, 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)-1. The Maple program is based on this recursive formula.

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).

A357702 Path length (total depths of vertices) of the rooted binary tree with Colijn-Plazzotta tree number n.

Original entry on oeis.org

0, 2, 6, 10, 12, 16, 22, 18, 22, 28, 34, 20, 24, 30, 36, 38, 26, 30, 36, 42, 44, 50, 34, 38, 44, 50, 52, 58, 66, 28, 32, 38, 44, 46, 52, 60, 54, 34, 38, 44, 50, 52, 58, 66, 60, 66, 42, 46, 52, 58, 60, 66, 74, 68, 74, 82, 50, 54, 60, 66, 68, 74, 82, 76, 82, 90

Offset: 1

Kevin Ryde, Oct 11 2022

In a rooted binary tree each vertex has 0 or 2 children.

All terms are even since each pair of 2 child vertices are at the same depth.

			For n=3, tree number 3 and the depth of each of its vertices is
      0  root
     / \
    1   1      total depths
   / \         a(3) = 0 + 1+1 + 2+2 = 6
  2   2

Kevin Ryde, PARI/GP Code

Cf. A357701 (vertex depths), A064002 (number of vertices).
Cf. A002024, A002260.
Cf. A196047 (in Matula-Goebel).

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

a(n) = a(x) + a(y) + A064002(n) - 1, for n>=2, where x = A002024(n-1) and y = A002260(n-1).

A198325 Irregular triangle read by rows: T(n,k) is the number of directed paths of length k (k>=1) in the rooted tree having Matula-Goebel number n (n>=2).

Original entry on oeis.org

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

Offset: 2

Emeric Deutsch, Nov 02 2011

A directed path of length k in a rooted tree is a sequence of k+1 vertices v[1], v[2], ..., v[k], v[k+1], such that v[j] is a child of v[j-1] for j = 2,3,...,k+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.
Number of entries in row n is A109082(n) (n=2,3,...).
Sum of entries in row n is A196047(n).
Sum(k*T(n,k),k>=1)=A198326(n).

			T(7,1)=3 and T(7,2)=2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having 3 directed paths of length 1 (the edges) and 2 directed paths of length 2.
Triangle starts:
  1;
  2,1;
  2;
  3,2,1;
  3,1;
  3,2;
  3;
  ...

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.

Cf. A109082, A196047, A196050, A198326.

with(numtheory): P := proc (n) local r, s, E: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: E := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+E(pi(n)) else E(r(n))+E(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*E(n)+x*P(pi(n)))) else sort(P(r(n)) +P(s(n))) end if end proc: T := proc (n, k) options operator, arrow: coeff(P(n), x, k) end proc: for n from 2 to 15 do seq(T(n, k), k = 1 .. degree(P(n))) end do; # yields sequence in triangular form
P(987654321); # yields P(987654321)

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
e[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, 1 + e[PrimePi[n]], True, e[r[n]] + e[s[n]]];
P[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, x*e[n] + x*P[PrimePi[n]], True, P[r[n]] + P[s[n]]];
T[n_] := Rest@CoefficientList[P[n], x];
Table[T[n], {n, 2, 50}] // Flatten (* Jean-François Alcover, Jun 22 2024, after Maple code *)

We give the recursive construction of the row generating polynomials P(n)=P(n,x): P(1)=0; if n=prime(t), then P(n)=x*E(n)+x*P(t), where E denotes number of edges (computed recursively and programmed in A196050); if n=r*s (r,s>=2), then P(n)=P(r)+P(s) (2nd Maple program yields P(n)).

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A061775 Number of nodes in rooted tree with Matula-Goebel number n.

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A196048 External path length of the rooted tree with Matula-Goebel number n.

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A196051 The Wiener index of the rooted tree with Matula-Goebel number n.

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A196068 Visitation length of the rooted tree with Matula-Goebel number n.

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

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A357702 Path length (total depths of vertices) of the rooted binary tree with Colijn-Plazzotta tree number n.

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