OFFSET
1,5
COMMENTS
This is the pruning operation mentioned, for example, in the Balaban reference (p. 360) and in the Todeschini - Consonni reference (p. 42).
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
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.
FORMULA
Let b(n)=A198328(n) (= the Matula-Goebel number of the tree obtained by removing the leaves together with their incident edges from the rooted tree with Matula-Goebel number n). a(1)=1; if n = prime(t), then a(n)=b(t); if n=r*s (r,s>=2), then a(n)=b(r)*b(s). The Maple program is based on this recursive formula.
EXAMPLE
a(7)=1 because the rooted tree with Matula-Goebel number 7 is Y; after deleting the vertices of degree one and their incident edges we obtain the 1-vertex tree having Matula-Goebel number 1.
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 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: seq(a(n), n = 1 .. 120);
MATHEMATICA
b[1] = b[2] = 1; b[n_] := b[n] = If[PrimeQ[n], Prime[b[PrimePi[n]]], Times @@ (b[#[[1]]]^#[[2]]& /@ FactorInteger[n])];
a[1] = 1; a[n_] := a[n] = If[PrimeQ[n], b[PrimePi[n]], Times @@ (b[#[[1]] ]^#[[2]]& /@ FactorInteger[n])];
Array[a, 100] (* Jean-François Alcover, Dec 18 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 24 2011
STATUS
approved