

A214567


Maximal number of distinct rooted trees obtained from the rooted tree with MatulaGoebel number n by adding one pendant edge at one of its vertices.


2



1, 2, 3, 2, 4, 4, 3, 2, 3, 5, 5, 4, 5, 4, 6, 2, 4, 4, 3, 5, 5, 6, 4, 4, 4, 6, 3, 4, 6, 7, 6, 2, 7, 5, 6, 4, 5, 4, 7, 5, 6, 6, 5, 6, 6, 5, 7, 4, 3, 5, 6, 6, 3, 4, 8, 4, 5, 7, 5, 7, 5, 7, 5, 2, 8, 8, 4, 5, 6, 7, 6, 4, 6, 6, 6, 4, 7, 8, 7, 5, 3
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OFFSET

1,2


COMMENTS

The MatulaGoebel number of a rooted tree can be 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.
Sum_{j such that rooted tree with MatulaGoebel number j has n vertices} a(j) = A000107(n). Example: the MatulaGoebel numbers of the rooted trees with 4 vertices are 5,6,7,8 and a(5)+a(6)+a(7)+a(8) = 4+4+3+2=13 = A000107(4).


REFERENCES

F. Goebel, On a 11 correspondence 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 YN. 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, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=1; if n is tth prime, then a(n)=1+a(t); if n is composite, then a(n) = 1+Sum_{pn}(a(p)1), where summation is over the distinct prime divisors of n.


EXAMPLE

a(4)=2 because the rooted tree with MatulaGoebel number 4 is V; adding an edge at either of the two leaves yields the same rooted tree.
a(5)=4 because the rooted tree with MatulaGoebel number 5 is the path on 4 vertices; adding one edge at any of the vertices yields a new rooted tree.
a(987654321)=18 (reader may verify this on Fig. 2 of the Deutsch paper).


MAPLE

with(numtheory): a := proc (n) local FS: FS := proc (n) options operator, arrow: factorset(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+a(pi(n)) else 1+add(a(FS(n)[j])1, j = 1 .. nops(FS(n))) end if end proc: seq(a(n), n = 1 .. 130);


PROG

(Haskell)
import Data.List (genericIndex)
a214567 n = genericIndex a214567_list (n  1)
a214567_list = 1 : g 2 where
g x = y : g (x + 1) where
y  t > 0 = a214567 t + 1
 otherwise = 1 + sum (map ((subtract 1) . a214567) $ a027748_row x)
where t = a049084 x
 Reinhard Zumkeller, Sep 03 2013


CROSSREFS

Cf. A000107.
Cf. A049084, a027748.
Sequence in context: A214906 A274228 A321476 * A259363 A087437 A304739
Adjacent sequences: A214564 A214565 A214566 * A214568 A214569 A214570


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Jul 25 2012


STATUS

approved



