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

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, 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, 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

Offset: 1

Author

Emeric Deutsch, Oct 17 2011

Keywords

• nonn

Comments

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.

Examples

			a(7)=0 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with all leaves at level 2.
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.
		

References

• F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
• I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
• I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
• D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.

Links

• Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A184154.

Programs

Maple

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);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
P[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, x*P[PrimePi[n]], True, P[r[n]] + P[s[n]]];
a[n_] := Exponent[Numerator[Together[P[n] /. x -> 1/x]], x];
Table[a[n], {n, 1, 110}] (* Jean-François Alcover, Jun 21 2024, after Maple code *)

Formula

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).