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%I #12 Apr 25 2021 22:15:49
%S 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,
%T 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,
%U 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
%N Internal path length of the rooted tree with Matula-Goebel number n.
%C The internal path length of a rooted tree is defined as the sum of the distances of all internal nodes to the root.
%C 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 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%H 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 D. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(n) = A196047(n) - A196048(n).
%F a(r*s) = a(r) + a(s).
%e a(7) = 1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (0+1).
%e 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.
%e 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.
%o (PARI) InIpl(n)={ if(n==1, return([0,0]),
%o my(f=factor(n)~, v=Mat(vector(#f,k,InIpl(primepi(f[1,k]))~)) );
%o 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]) ] ) )
%o };
%o A343006(n) = InIpl(n)[2];
%Y Cf. A196047, A196048.
%K nonn
%O 1,5
%A _François Marques_, Apr 02 2021