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A343006
Internal path length of the rooted tree with Matula-Goebel number n.
2
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, 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, 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
OFFSET
1,5
COMMENTS
The internal path length of a rooted tree is defined as the sum of the distances of all internal nodes to the root.
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.
FORMULA
a(n) = A196047(n) - A196048(n).
a(r*s) = a(r) + a(s).
EXAMPLE
a(7) = 1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (0+1).
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.
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.
PROG
(PARI) InIpl(n)={ if(n==1, return([0, 0]),
my(f=factor(n)~, v=Mat(vector(#f, k, InIpl(primepi(f[1, k]))~)) );
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]) ] ) )
};
A343006(n) = InIpl(n)[2];
CROSSREFS
Sequence in context: A011354 A143119 A220419 * A085565 A216677 A341412
KEYWORD
nonn
AUTHOR
François Marques, Apr 02 2021
STATUS
approved