

A198329


The MatulaGoebel number of the rooted tree obtained from the rooted tree with MatulaGoebel 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
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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, 355375, 1979.
R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, WileyVCH, 2000.


LINKS

Table of n, a(n) for n=1..83.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Let b(n)=A198328(n) (= the MatulaGoebel number of the tree obtained by removing the leaves together with their incident edges from the rooted tree with MatulaGoebel number n). a(1)=1; if n = p(t) (=the tth 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 MatulaGoebel number 7 is Y; after deleting the vertices of degree one and their incident edges we obtain the 1vertex tree having MatulaGoebel 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] (* JeanFrançois 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



