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

%I A196068 #26 Jun 25 2024 18:05:50
%S A196068 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,
%T A196068 15,16,13,18,17,21,11,20,16,18,15,16,14,18,16,19,16,17,19,20,19,22,14,
%U A196068 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
%N A196068 Visitation length of the rooted tree with Matula-Goebel number n.
%C A196068 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).
%C A196068 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.
%D A196068 F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A196068 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A196068 I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A196068 M. Keijzer and J. Foster, Crossover bias in genetic programming, Lecture Notes in Computer Sciences, 4445, 2007, 33-44.
%D A196068 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A196068 Reinhard Zumkeller, <a href="/A196068/b196068.txt">Table of n, a(n) for n = 1..10000</a>
%H A196068 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 A196068 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A196068 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.
%e A196068 a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+2+2+4=9).
%e A196068 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).
%p A196068 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);
%t A196068 r[n_] := FactorInteger[n][[1, 1]];
%t A196068 s[n_] := n/r[n];
%t A196068 nn[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + nn[PrimePi[n]], True, nn[r[n]] + nn[s[n]] - 1];
%t A196068 a[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, a[PrimePi[n]] + nn[PrimePi[n]] + 1, True, a[r[n]] + a[s[n]] - 1];
%t A196068 Table[a[n], {n, 1, 80}] (* _Jean-François Alcover_, Jun 25 2024, after Maple code *)
%o A196068 (Haskell)
%o A196068 import Data.List (genericIndex)
%o A196068 a196068 n = genericIndex a196068_list (n - 1)
%o A196068 a196068_list = 1 : g 2 where
%o A196068    g x = y : g (x + 1) where
%o A196068      y | t > 0     = a196068 t + a061775 t + 1
%o A196068        | otherwise = a196068 r + a196068 s - 1
%o A196068        where t = a049084 x; r = a020639 x; s = x `div` r
%o A196068 -- _Reinhard Zumkeller_, Sep 03 2013
%Y A196068 Cf. A196046, A196047.
%Y A196068 Cf. A049084, A020639, A061775.
%K A196068 nonn
%O A196068 1,2
%A A196068 _Emeric Deutsch_, Oct 04 2011