

A184166


The level of the lowest leaf of the rooted tree with MatulaGoebel number n.


1



0, 1, 2, 1, 3, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 5, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1
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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

Table of n, a(n) for n=1..110.
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 is the tth prime, then a(n) = 1 + a(t); if n is composite, n=rs, then a(n) = min(a(r),a(s)). The Maple program is based on this recursive rule.


EXAMPLE

a(7)=2 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y (all leaves are at level 2).


MAPLE

with (numtheory):
a:= proc(n) option remember;
if n=1 then 0
elif isprime(n)=true then 1 +a(pi(n))
else min (seq (a(i), i=factorset(n)))
fi
end:
seq (a(n), n=1..200);


CROSSREFS

Cf. A184167.
Sequence in context: A185894 A214180 A343073 * A029423 A184169 A176207
Adjacent sequences: A184163 A184164 A184165 * A184167 A184168 A184169


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 22 2011


STATUS

approved



