A214573 -id:A214573 - 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

A214574 The Strahler number of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3

Offset: 1

Emeric Deutsch, Aug 14 2012

nonn

The Strahler number of a vertex of a rooted tree is defined recursively in the following way: (i) the Strahler number of a leaf is 1; (ii) if the vertex has one child with Strahler number i and all other children have Strahler number less than i, then the Strahler number of the vertex is again i; (iii) if the vertex has two or more children with Strahler number i and no child with Strahler number greater than i, then the Strahler number of the vertex is i+1. See the Wikipedia reference. The Strahler number of a rooted tree T is defined as the Strahler number of the root of T.

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(4)=2 because the rooted tree with Matula-Goebel number 4 is V; the two leaves have Strahler numbers 1,1, and the root has Strahler number 2; this is - by definition - the Strahler number of the tree.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
Emeric Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
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.
Wikipedia, Strahler number
Index entries for sequences related to Matula-Goebel numbers

Cf. A061775, A214573.

Cf. A356082 (most likely the positions of the first occurrences of each n, also positions of the records).

with(numtheory): G := 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 x elif bigomega(n) = 1 then sort(expand(x^degree(G(pi(n)))+G(pi(n)))) elif 1 < bigomega(n) and degree(G(r(n))) <> degree(G(s(n))) then sort(G(r(n))-x^degree(G(r(n)))+G(s(n))-x^degree(G(s(n)))+x^max(degree(G(r(n))), degree(G(s(n))))) else sort(G(r(n))-x^degree(G(r(n)))+G(s(n))-x^degree(G(s(n)))+x^(1+degree(G(r(n))))) end if end proc: seq(degree(G(n)), n = 1 .. 200);

A214574(n) = if(1==n, 1, if(isprime(n), A214574(primepi(n)), my(f=factor(n), m, mpi=0); f[,1]=apply(A214574,f[,1]); m = vecmax(f[,1]); for(i=1,#f~,if(m==f[i,1], if(mpi || f[i,2]>1, return(1+m), mpi = i))); (m))); \\ Antti Karttunen, Jan 20 2025

Define the Strahler polynomial of a rooted tree T as the generating polynomial of the vertices of T with respect to their Strahler numbers. For example, it follows at once that the Strahler polynomial of the rooted tree V is 2x + x^2. Denote by G(n)=G(n;x) the Strahler polynomial of the rooted tree with Matula-Goebel number n. Clearly, A214573(n,k) is the coefficient of x^k in G(n). We have (i) G(1)= x; (ii) if n=p(t) (the t-th prime), then G(n) = x^{degree(G(t)} + G(t); (iii) if n=rs (r,s>=2), then G(n) = G(r) - degree (G(r)) + G(s) - degree(G(s) + x^m, where m = 1+degree(G(r)) if degree(G(r))=degree(G(s)) and m = max(degree(G(r), G(s)) otherwise. The Strahler number a(n) = degree(G(n)).

cp's OEIS Frontend

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

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