A190175 - OEIS

%I #25 Jun 21 2024 07:34:34

%S 2,7,8,28,32,43,53,98,112,128,172,212,227,263,311,343,392,443,448,512,

%T 577,602,688,742,848,908,1052,1193,1244,1372,1423,1568,1619,1772,1792,

%U 1993,2048,2107,2308,2311,2408,2539,2597,2752,2939,2968,3178,3209,3392,3632,3682,3698,3779

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

%C 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.

%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.

%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.

%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.

%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.

%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

%F 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.

%F 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).

%e 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.

%e 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.

%p 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;

%t r[n_] := FactorInteger[n][[1, 1]];

%t s[n_] := n/r[n];

%t 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]];

%t a[n_] := (1/2)(g[n] /. x -> 1) + (1/2)(g[n] /. x -> -1);

%t A = {};

%t Do[If[a[n] == 0, A = Union[A, {n}]], {n, 1, 4000}];

%t A (* _Jean-François Alcover_, Jun 20 2024, after Maple code *)

%Y Cf. A182907, A190174.

%K nonn

%O 1,1

%A _Emeric Deutsch_, Oct 30 2011, Dec 09 2011