

A196062


Number of leafparents of the rooted tree with MatulaGoebel number n.


2



0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 2, 1, 1, 3, 1, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 1, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 2, 3, 1, 4, 2, 2, 2, 3, 1, 3, 3, 2, 3, 1, 3, 3, 1, 2, 3, 3, 2, 3, 2, 3, 3, 2, 2, 4, 2, 2, 4, 3, 2, 3, 2, 3, 3, 2, 2, 4, 3, 3, 2, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 4, 2, 3
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OFFSET

1,6


COMMENTS

A leafparent in a rooted tree is a node that is the parent of at least one leaf.
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; a(2)=1; if n=p(t) (the tth prime, t>1), then a(n)=a(t); if n=rs (r,s >=2) and both r and s are even, then a(n)=a(r)+a(s)1; if n=rs (r,s >=2) and not both r and s are even, then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.


EXAMPLE

a(7)=1 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y.


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 n = 2 then 1 elif bigomega(n) = 1 then a(pi(n)) elif `mod`(r(n), 2) = 0 and `mod`(s(n), 2) = 0 then a(r(n))+a(s(n))1 else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);


PROG

(Haskell)
import Data.List (genericIndex)
a196062 n = genericIndex a196062_list (n  1)
a196062_list = 0 : 1 : g 3 where
g x = y : g (x + 1) where
y  t > 0 = a196062 t
 otherwise = a196062 r + a196062 s  0 ^ (x `mod` 4)
where t = a049084 x; r = a020639 x; s = x `div` r
 Reinhard Zumkeller, Sep 03 2013


CROSSREFS

Cf. A049084, A020639.
Sequence in context: A074265 A254688 A002636 * A283682 A087974 A008679
Adjacent sequences: A196059 A196060 A196061 * A196063 A196064 A196065


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Sep 30 2011


STATUS

approved



