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A184159 The difference between the levels of the highest and lowest leaves in the rooted tree with Matula-Goebel number n.

%I A184159 #18 Jun 21 2024 15:38:41
%S A184159 0,0,0,0,0,1,0,0,0,2,0,1,1,1,1,0,0,1,0,2,0,3,0,1,0,2,0,1,2,2,0,0,2,2,
%T A184159 1,1,1,1,1,2,1,1,1,3,1,2,1,1,0,2,1,2,0,1,1,1,0,3,0,2,1,4,0,0,1,3,0,2,
%U A184159 1,2,2,1,0,2,1,1,2,2,3,2,0,3,0,1,0,2,2,3,1,2,1,2,3,3,1,1,0,1,2,2,2,2,0,2,1,1,1,1,2,3
%N A184159 The difference between the levels of the highest and lowest leaves in the rooted tree with Matula-Goebel number n.
%C A184159 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 A184159 F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D A184159 I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D A184159 I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D A184159 D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H A184159 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%H A184159 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A184159 In A184154 one constructs for each n the generating polynomial P(n,x) of the leaves of the rooted tree with Matula-Goebel number n, according to their levels. a(n) = degree of the numerator of P(n,1/x) (see the Maple program).
%e A184159 a(7)=0 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with all leaves at level 2.
%e A184159 a(2^m)=0 because the rooted tree with Matula-Goebel number 2^m is the star with m edges; all leaves are at level 1.
%p A184159 with(numtheory): a := proc (n) local r, s, P: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: P := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: degree(numer(subs(x = 1/x, P(n)))) end proc; seq(a(n), n = 1 .. 110);
%t A184159 r[n_] := FactorInteger[n][[1, 1]];
%t A184159 s[n_] := n/r[n];
%t A184159 P[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, x*P[PrimePi[n]], True, P[r[n]] + P[s[n]]];
%t A184159 a[n_] := Exponent[Numerator[Together[P[n] /. x -> 1/x]], x];
%t A184159 Table[a[n], {n, 1, 110}] (* _Jean-François Alcover_, Jun 21 2024, after Maple code *)
%Y A184159 Cf. A184154.
%K A184159 nonn
%O A184159 1,10
%A A184159 _Emeric Deutsch_, Oct 17 2011