A198337 Radius of rooted tree having Matula-Goebel number n.
0, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 1, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 3, 1, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 3, 2, 3, 2, 2, 3, 3, 2, 1, 2, 4, 2, 2, 3, 2, 3, 2, 3, 2, 1, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 3, 2, 3, 3, 3, 2, 2, 3, 2, 2, 3, 2
Offset: 1
Keywords
Examples
a(7)=1 because the rooted tree with Matula-Goebel number 7 is Y and its vertices have eccentricities 2,2,2,1. a(11)=2 because the rooted tree with Matula-Goebel number 11 is the path tree on 5 vertices and the eccentricities are 4,4,3,3,2.
References
- A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
- 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.
Crossrefs
Cf. A198336.
Programs
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Maple
with(numtheory): aa := 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 n = 2 then 1 elif bigomega(n) = 1 then ithprime(b(pi(n))) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then b(pi(n)) else b(r(n))*b(s(n)) end if end proc: S := proc (m) local A, i: A[m, 1] := m; for i while aa(A[m, i]) < A[m, i] do A[m, i+1] := aa(A[m, i]) end do: seq(A[m, j], j = 1 .. i) end proc; a := proc (n) options operator, arrow: nops([S(n)])-1 end proc: seq(a(n), n = 1 .. 110);
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Mathematica
r[n_] := FactorInteger[n][[1, 1]]; s[n_] := n/r[n]; b[n_] := Which[n == 1, 1, n == 2, 1, PrimeOmega[n] == 1, Prime[b[PrimePi[n]]], True, b[r[n]]*b[s[n]]]; aa[n_] := Which[n == 1, 1, PrimeOmega[n] == 1, b[PrimePi[n]], True, b[r[n]]*b[s[n]]]; S[m_] := Module[{A, i}, A[m, 1] = m; For[i = 1, aa[A[m, i]] < A[m, i], i++, A[m, i + 1] = aa[A[m, i]]]; Table[A[m, j], {j, 1, i}]]; a[n_] := Length[S[n]] - 1; Table[a[n], {n, 1, 110}] (* Jean-François Alcover, Aug 12 2024, after Emeric Deutsch *)
Formula
A198336(n) gives the sequence of the Matula-Goebel numbers of the rooted trees obtained from the rooted tree with Matula-Goebel number n by pruning it successively 0,1,2,... times. Then the radius of the rooted tree with Matula-Goebel number n is equal to the number of terms in this sequence diminished by 1.
Comments