

A196047


Path length of the rooted tree with MatulaGoebel number n.


6



0, 1, 3, 2, 6, 4, 5, 3, 6, 7, 10, 5, 8, 6, 9, 4, 9, 7, 7, 8, 8, 11, 11, 6, 12, 9, 9, 7, 12, 10, 15, 5, 13, 10, 11, 8, 10, 8, 11, 9, 13, 9, 11, 12, 12, 12, 15, 7, 10, 13, 12, 10, 9, 10, 16, 8, 10, 13, 14, 11, 13, 16, 11, 6, 14, 14, 12, 11, 14, 12, 14, 9, 14, 11, 15, 9, 15, 12, 17, 10, 12, 14, 17, 10, 15, 12, 15, 13, 12, 13, 13, 13, 18, 16, 13, 8, 19, 11, 16, 14
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OFFSET

1,3


COMMENTS

The path length of a rooted tree is defined as the sum of distances of all nodes to the root of the tree.
The MatulaGoebel number of a rooted tree is defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


REFERENCES

F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=0; if n=p(t) (= the tth prime) then a(n)=a(t)+N(t), where N(t) is the number of nodes of the rooted tree with Matula number t; if n=rs (r,s>=2), then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.


EXAMPLE

a(7)=5 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y (1+2+2=5).
a(2^m) = m because the rooted tree with MatulaGoebel number 2^m is a star with m edges.


MAPLE

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 0 elif bigomega(n) = 1 then a(pi(n))+N(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 100);


PROG

(Haskell)
import Data.List (genericIndex)
a196047 n = genericIndex a196047_list (n  1)
a196047_list = 0 : g 2 where
g x = y : g (x + 1) where
y = if t > 0 then a196047 t + a061775 t else a196047 r + a196047 s
where t = a049084 x; r = a020639 x; s = x `div` r
 Reinhard Zumkeller, Sep 03 2013


CROSSREFS

Cf. A049084, A020639.
Sequence in context: A121647 A033940 A286367 * A106409 A115510 A230598
Adjacent sequences: A196044 A196045 A196046 * A196048 A196049 A196050


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sep 27 2011


STATUS

approved



