cp's OEIS Frontend

A196049 Number of branching nodes of the rooted tree with Matula-Goebel number n.

%I A196049 #28 Jun 26 2024 17:19:44
%S A196049 0,0,0,0,0,0,1,1,0,0,0,1,1,1,0,1,1,1,1,1,1,0,1,1,0,1,1,2,1,1,0,1,0,1,
%T A196049 1,1,1,1,1,1,1,2,2,1,1,1,1,1,2,1,1,2,1,1,0,2,1,1,1,1,1,0,2,1,1,1,1,2,
%U A196049 1,2,1,1,2,1,1,2,1,2,1,1,1,1,1,2,1,2,1,1,1,1,2,2,0,1,1,1,1,3,1,1,2,2,1,2,2,1,2,1,1,1
%N A196049 Number of branching nodes of the rooted tree with Matula-Goebel number n.
%C A196049 A branching node of a tree is a vertex of degree at least 3.
%C A196049 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.
%H A196049 Reinhard Zumkeller, <a href="/A196049/b196049.txt">Table of n, a(n) for n = 1..10000</a>
%H A196049 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%H A196049 F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H A196049 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A196049 I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H A196049 D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H A196049 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A196049 a(1)=0; if n=prime(t) and t is not the product of 2 prime factors, then a(n)=a(t); if n=prime(t) and t is the product of 2 prime factors, then a(n)=a(t)+1; if n=r*s (r prime, s>=2) and s is not a product of 2 prime factors, then a(n)=a(r)+a(s); if n=r*s (r prime, s>=2) and s is a product of 2 prime factors, then a(n)=a(r)+a(s)+1. The Maple program is based on this recursive formula.
%e A196049 a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
%e A196049 if m>2 then a(2^m) = 1 because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%p A196049 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 bigomega(n) = 1 and bigomega(pi(n)) <> 2 then a(pi(n)) elif bigomega(n) = 1 then a(pi(n))+1 elif bigomega(s(n)) <> 2 then a(r(n))+a(s(n)) else a(r(n))+a(s(n))+1 end if end proc: seq(a(n), n = 1 .. 110);
%t A196049 r[n_] := FactorInteger[n][[1, 1]];
%t A196049 s[n_] := n/r[n];
%t A196049 a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] != 2, a[PrimePi[n]], PrimeOmega[n] == 1, a[PrimePi[n]] + 1, PrimeOmega[s[n]] != 2, a[r[n]] + a[s[n]], True, a[r[n]] + a[s[n]] + 1];
%t A196049 Table[a[n], {n, 1, 110}] (* _Jean-François Alcover_, Jun 25 2024, after Maple code *)
%o A196049 (Haskell)
%o A196049 import Data.List (genericIndex)
%o A196049 a196049 n = genericIndex a196049_list (n - 1)
%o A196049 a196049_list = 0 : g 2 where
%o A196049    g x = y : g (x + 1) where
%o A196049      y | t > 0     = a196049 t + a064911 t
%o A196049        | otherwise = a196049 r + a196049 s + a064911 s
%o A196049        where t = a049084 x; r = a020639 x; s = x `div` r
%o A196049 -- _Reinhard Zumkeller_, Sep 03 2013
%Y A196049 Cf. A049084, A020639, A064911.
%K A196049 nonn
%O A196049 1,28
%A A196049 _Emeric Deutsch_, Sep 27 2011