

A196046


Maximum vertexdegree in the rooted tree with MatulaGoebel number n.


3



0, 1, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 2, 4, 3, 3, 4, 3, 3, 2, 3, 4, 2, 3, 3, 3, 3, 3, 2, 5, 2, 3, 3, 4, 4, 4, 3, 4, 3, 3, 3, 3, 3, 3, 3, 5, 3, 3, 3, 3, 5, 4, 2, 4, 4, 3, 3, 4, 4, 2, 3, 6, 3, 3, 4, 3, 3, 3, 4, 5, 3, 4, 3, 4, 3, 3, 3, 5, 4, 3, 3, 4, 3, 3, 3, 4, 5, 4, 3, 3, 2, 3, 4, 6, 3, 3, 3, 4, 3, 3, 4, 4, 3, 5, 4, 5, 3, 3
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OFFSET

1,3


COMMENTS

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) = max(a(t), 1+G(t)); if n=rs (r,s>=2), then a(n)=max(a(r),a(s), G(r)+G(s)); G(m) is the number of prime divisors of m counted with multiplicity. The Maple program is based on this recursive formula.
The Gutman et al. references contain a different recursive formula.


EXAMPLE

a(7)=3 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y.
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: 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 then max(a(pi(n)), 1+bigomega(pi(n))) else max(a(r(n)), a(s(n)), bigomega(r(n))+bigomega(s(n))) end if end proc: seq(a(n), n = 1 .. 110);


PROG

(Haskell)
import Data.List (genericIndex)
a196046 n = genericIndex a196046_list (n  1)
a196046_list = 0 : g 2 where
g x = y : g (x + 1) where
y  t > 0 = max (a196046 t) (a001222 t + 1)
 otherwise = maximum [a196046 r, a196046 s, a001222 r + a001222 s]
where t = a049084 x; r = a020639 x; s = x `div` r
 Reinhard Zumkeller, Sep 03 2013


CROSSREFS

Cf. A049084, A020639, A001222.
Sequence in context: A322665 A273632 A347387 * A152724 A081743 A247069
Adjacent sequences: A196043 A196044 A196045 * A196047 A196048 A196049


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sep 26 2011


STATUS

approved



