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%I #15 Mar 07 2017 11:32:03
%S 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,
%T 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,
%U 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
%N The level of the lowest leaf of the rooted tree with Matula-Goebel number n.
%C The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel 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 Matula-Goebel numbers of the m branches of T.
%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288, 2011
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(1)=0; if n is the t-th 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.
%e a(7)=2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (all leaves are at level 2).
%p with (numtheory):
%p a:= proc(n) option remember;
%p if n=1 then 0
%p elif isprime(n)=true then 1 +a(pi(n))
%p else min (seq (a(i), i=factorset(n)))
%p fi
%p end:
%p seq (a(n), n=1..200);
%Y Cf. A184167.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Oct 22 2011
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