A214572 -id:A214572 - 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

A061773 Triangle in which n-th row lists Matula-Goebel numbers for all rooted trees with n nodes.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 15, 18, 20, 21, 22, 23, 24, 26, 28, 29, 31, 32, 34, 37, 38, 41, 43, 53, 59, 67, 25, 27, 30, 33, 35, 36, 39, 40, 42, 44, 46, 47, 48, 49, 51, 52, 56, 57, 58, 61, 62, 64, 68, 71, 73, 74, 76, 79, 82, 83, 86, 89, 101, 106

Offset: 1

N. J. A. Sloane, Jun 22 2001

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.
n-th row has A000081(n) terms.
First entry in row n is A005517(n).
Last entry in row n is A005518(n).
The Maple program yields row n after defining F = A005517(n) and L = A005518(n).

			The labels for the rooted trees with at most 4 nodes are as follows (x is the root):
                                         o
                                         |
               o         o        o   o  o
               |          \        \ /   |
     o  o   o  o  o o o    o   o    o    o
     |   \ /   |   \|/      \ /     |    |
  x  x    x    x    x        x      x    x
  1  2    4    3    8        6      7    5 (label)
Triangle begins:
1;
2;
3,4;
5,6,7,8;
9,10,11,12,13,14,16,17,19;
15,18,20,21,22,23,24,26,28,29,31,32,34,37,38,41,43,53,59,67;
25,27,30,33,35,36,39,40,42,44,46,47,48,49,51,52,56,57,58,61,62,64,68,\
71,73,74,76,79,82,83,86,89,101,106,107,109,118,127,131,134,139,157,163,\
179,191,241,277,331;
...
Triangle of rooted trees represented as finitary multisets begins:
(),
(()),
((())), (()()),
(((()))), (()(())), ((()())), (()()()),
((())(())), (()((()))), ((((())))), (()()(())), ((()(()))), (()(()())), (()()()()), (((()()))), ((()()())). - _Gus Wiseman_, Dec 21 2016

Alois P. Heinz, Rows n = 1..12, flattened
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.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Kevin Ryde, PARI/GP code calculating rows
Gus Wiseman, Matula-Goebel trees triangle illustration n=1..6
Index entries for sequences related to Matula-Goebel numbers
Index entries for sequences that are permutations of the natural numbers
Index entries for sequences related to rooted trees
Index entries for sequences related to trees

Cf. A061775 (number of nodes), A000081 (row lengths), A005517 (row minimum), A005518 (row maximum), A214572 (row n=8).
Cf. A347620 (inverse permutation).
Cf. A276625, A279065, A279614.

n := 8: F := 45: L := 2221: with(numtheory): N := proc (m) local r, s: r := proc (m) options operator, arrow: op(1, factorset(m)) end proc: s := proc (m) options operator, arrow: m/r(m) end proc: if m = 1 then 1 elif bigomega(m) = 1 then 1+N(pi(m)) else N(r(m))+N(s(m))-1 end if end proc: A := {}: for k from F to L do if N(k) = n then A := `union`(A, {k}) else  end if end do: A;

F[n_] := F[n] = Which[n == 1, 1, n == 2, 2, Mod[n, 3] == 0, 3*5^(n/3-1), Mod[n, 3] == 1, 5^(n/3-1/3), True, 9*5^(n/3-5/3)]; L[n_] := L[n] = Switch[n, 1, 1, 2, 2, 3, 4, 4, 8, , Prime[L[n-1]]]; r[m] := FactorInteger[m][[1, 1]]; s[m_] := m/r[m]; NN[m_] := NN[m] = Which[m == 1, 1, PrimeOmega[m] == 1, 1+NN[PrimePi[m]], True, NN[r[m]]+NN[s[m]]-1]; row[n_] := Module[{A, k}, A = {}; For[k = F[n], k <= L[n], k++, If[NN[k] == n, A = Union[A, {k}]]]; A]; Table[row[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Mar 06 2014, after Maple *)
nn=8;MGweight[n_]:=If[n===1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>k*MGweight[PrimePi[p]]]]];
Take[GatherBy[Range[Switch[nn,1,1,2,2,3,4,,Nest[Prime,8,nn-4]]],MGweight],nn] (* _Gus Wiseman, Dec 21 2016 *)

PARI
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\\ See links.
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More terms from Emeric Deutsch, May 01 2004

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

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