A184161 Number of subtrees in the rooted tree with Matula-Goebel number n.

1, 3, 6, 6, 10, 10, 11, 11, 15, 15, 15, 17, 17, 17, 21, 20, 17, 25, 20, 24, 24, 21, 25, 30, 28, 25, 36, 28, 24, 34, 21, 37, 28, 24, 32, 44, 30, 30, 34, 41, 25, 40, 28, 32, 48, 36, 34, 55, 37, 45, 32, 40, 37, 64, 36, 49, 41, 34, 24, 59, 44, 28, 57, 70, 44, 44, 30, 37, 48, 53, 41, 81, 40, 44, 63, 49, 41, 56, 32, 74

Offset: 1

Author

Emeric Deutsch, Oct 19 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

			a(4) = 6 because the rooted tree with Matula-Goebel number 4 is V; it has 6 subtrees (three 1-vertex subtrees, two 1-edge subtrees, and the tree itself).
		

Links

• É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
• Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint 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. A184160 (subtrees containing the root), A184164 (numbers not occurring as terms).

Programs

Maple

with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+b(pi(n)) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n))+b(pi(n))+1 else a(r(n))+a(s(n))+(b(r(n))-1)*(b(s(n))-1)-1 end if end proc: seq(a(n), n = 1 .. 80);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
b[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, 1 + b[PrimePi[n]], True, b[r[n]]*b[s[n]]];
a[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, a[PrimePi[n]] + b[PrimePi[n]] + 1, True, a[r[n]] + a[s[n]] + (b[r[n]] - 1)*(b[s[n]] - 1) - 1];
Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Jun 24 2024, after Maple code *)

Formula

Let b(n)=A184160(n) denote the number of those subtrees of the rooted tree with Matula-Goebel number n that contain the root. Then a(1)=1; if n=prime(t) (=the t=th prime), then a(n)=1+a(t)+b(t); if n=r*s (r,s,>=2), then a(n)=a(r)+a(s)+(b(r)-1)*(b(s)-1)-1. The Maple program is based on this recursive formula.