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 A198329 The Matula-Goebel number of the rooted tree obtained from the rooted tree with Matula-Goebel number n after removing the vertices of degree one, together with their incident edges. 5
 1, 1, 1, 1, 2, 2, 1, 1, 4, 3, 3, 2, 2, 2, 6, 1, 2, 4, 1, 3, 4, 5, 4, 2, 9, 3, 8, 2, 3, 6, 5, 1, 10, 3, 6, 4, 2, 2, 6, 3, 3, 4, 2, 5, 12, 7, 6, 2, 4, 9, 6, 3, 1, 8, 15, 2, 4, 5, 3, 6, 4, 11, 8, 1, 9, 10, 2, 3, 14, 6, 3, 4, 4, 3, 18, 2, 10, 6, 5, 3, 16, 5, 7 (list; graph; refs; listen; history; text; internal format)
 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 = p(t) (=the t-th prime), then a(n)=b(t); if n=rs (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 Cf. A198328. Sequence in context: A202735 A116595 A128315 * A123566 A279863 A201757 Adjacent sequences:  A198326 A198327 A198328 * A198330 A198331 A198332 KEYWORD nonn AUTHOR Emeric Deutsch, Nov 24 2011 STATUS approved

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Last modified January 23 03:08 EST 2019. Contains 319370 sequences. (Running on oeis4.)