A190175 The Goebel-Matula numbers of the rooted trees having only vertices of odd degree.

2, 7, 8, 28, 32, 43, 53, 98, 112, 128, 172, 212, 227, 263, 311, 343, 392, 443, 448, 512, 577, 602, 688, 742, 848, 908, 1052, 1193, 1244, 1372, 1423, 1568, 1619, 1772, 1792, 1993, 2048, 2107, 2308, 2311, 2408, 2539, 2597, 2752, 2939, 2968, 3178, 3209, 3392, 3632, 3682, 3698, 3779

Offset: 1

Author

Emeric Deutsch, Oct 30 2011, Dec 09 2011

Keywords

• nonn

Comments

The Matula-Goebel number of a rooted tree can be 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

			7 is in the sequence because the rooted tree with Matula-Goebel number 7 is the rooted tree Y with vertices of degree 1,1,1,3.
15 is not in the sequence because the rooted tree with Matula-Goebel number 15 is the path tree ABRCDE, rooted at R; it has 2 vertices of degree 1 and 4 vertices of degree 2.
		

Links

• Emeric Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
• 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 Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A182907, A190174.

Programs

Maple

with(numtheory): g := 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 1 elif bigomega(n) = 1 then sort(expand(g(pi(n))+x^bigomega(pi(n))*(x-1)+x)) else sort(expand(g(r(n))+g(s(n))-x^bigomega(r(n))-x^bigomega(s(n))+x^bigomega(n))) end if end proc: a := proc (n) options operator, arrow: (1/2)*subs(x = 1, g(n))+(1/2)*subs(x = -1, g(n)) end proc: A := {}: for n to 4000 do if a(n) = 0 then A := `union`(A, {n}) else  end if end do: A;

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
g[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, g[PrimePi[n]] + x^PrimeOmega[PrimePi[n]]*(x - 1) + x, True, g[r[n]] + g[s[n]] - x^PrimeOmega[r[n]] - x^PrimeOmega[s[n]] + x^PrimeOmega[n]];
a[n_] := (1/2)(g[n] /. x -> 1) + (1/2)(g[n] /. x -> -1);
A = {};
Do[If[a[n] == 0, A = Union[A, {n}]], {n, 1, 4000}];
A (* Jean-François Alcover, Jun 20 2024, after Maple code *)

Formula

The number of vertices of even degree of the rooted trees with Matula-Goebel number n is A190174(n). The number n is in the sequence if and only if A190174(n)=0.

In A182907 one can find the generating polynomial g(n)=g(n,x) of the vertices of the rooted tree having Matula-Goebel number n, according to degree. We look for those values of n for which the polynomial g(n,x) is odd, i.e. satisfies g(n,-x)=-g(n,x).