A198328 The Matula-Goebel number of the rooted tree obtained from the rooted tree with Matula-Goebel number n after removing the leaves, together with their incident edges.
1, 1, 2, 1, 3, 2, 2, 1, 4, 3, 5, 2, 3, 2, 6, 1, 3, 4, 2, 3, 4, 5, 7, 2, 9, 3, 8, 2, 5, 6, 11, 1, 10, 3, 6, 4, 3, 2, 6, 3, 5, 4, 3, 5, 12, 7, 13, 2, 4, 9, 6, 3, 2, 8, 15, 2, 4, 5, 5, 6, 7, 11, 8, 1, 9, 10, 3, 3, 14, 6, 5, 4, 7, 3, 18, 2, 10, 6, 11, 3, 16, 5
Offset: 1
Examples
a(7)=2 because the rooted tree with Matula-Goebel number 7 is Y; after deleting the leaves and their incident edges, we obtain the 1-edge tree having Matula-Goebel number 2.
References
- A. T. Balaban, Chemical graphs, Theoret. Chim. Acta (Berl.) 53, 355-375, 1979.
- R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, 2000.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
- 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. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
- Index entries for sequences related to Matula-Goebel numbers
Programs
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Haskell
import Data.List (genericIndex) a198328 n = genericIndex a198328_list (n - 1) a198328_list = 1 : 1 : g 3 where g x = y : g (x + 1) where y = if t > 0 then a000040 (a198328 t) else a198328 r * a198328 s where t = a049084 x; r = a020639 x; s = x `div` r -- Reinhard Zumkeller, Sep 03 2013 -
Maple
with(numtheory): a := 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 n = 2 then 1 elif bigomega(n) = 1 then ithprime(a(pi(n))) else a(r(n))*a(s(n)) end if end proc; seq(a(n), n = 1 .. 120);
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Mathematica
a[1] = a[2] = 1; a[n_] := a[n] = If[PrimeQ[n], Prime[a[PrimePi[n]]], Times @@ (a[#[[1]]]^#[[2]]& /@ FactorInteger[n])]; Array[a, 100] (* Jean-François Alcover, Dec 18 2017 *)
Formula
a(1)=1; a(2)=1; if n=prime(t) (t>1), then a(n)=prime(a(t)); if n=r*s (r,s,>=2), then a(n)=a(r)*a(s). The Maple program is based on this recursive formula.
Completely multiplicative with a(2) = 1, a(prime(t)) = prime(a(t)) for t > 1. - Andrew Howroyd, Aug 01 2018
Comments