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A214567 Maximal number of distinct rooted trees obtained from the rooted tree with Matula-Goebel number n by adding one pendant edge at one of its vertices.

%I A214567 #25 Jun 25 2024 13:11:15
%S A214567 1,2,3,2,4,4,3,2,3,5,5,4,5,4,6,2,4,4,3,5,5,6,4,4,4,6,3,4,6,7,6,2,7,5,
%T A214567 6,4,5,4,7,5,6,6,5,6,6,5,7,4,3,5,6,6,3,4,8,4,5,7,5,7,5,7,5,2,8,8,4,5,
%U A214567 6,7,6,4,6,6,6,4,7,8,7,5,3
%N A214567 Maximal number of distinct rooted trees obtained from the rooted tree with Matula-Goebel number n by adding one pendant edge at one of its vertices.
%C A214567 The Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
%C A214567 Sum_{j such that rooted tree with Matula-Goebel number j has n vertices} a(j) = A000107(n). Example: the Matula-Goebel numbers of the rooted trees with 4 vertices are 5,6,7,8 and a(5)+a(6)+a(7)+a(8) = 4+4+3+2=13 = A000107(4).
%H A214567 Reinhard Zumkeller, <a href="/A214567/b214567.txt">Table of n, a(n) for n = 1..10000</a>
%H A214567 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288"> Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H A214567 F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H A214567 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A214567 I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H A214567 D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H A214567 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A214567 a(1)=1; if n is t-th prime, then a(n)=1+a(t); if n is composite, then a(n) = 1+Sum_{p|n}(a(p)-1), where summation is over the distinct prime divisors of n.
%e A214567 a(4)=2 because the rooted tree with Matula-Goebel number 4 is V; adding an edge at either of the two leaves yields the same rooted tree.
%e A214567 a(5)=4 because the rooted tree with Matula-Goebel number 5 is the path on 4 vertices; adding one edge at any of the vertices yields a new rooted tree.
%e A214567 a(987654321)=18 (reader may verify this on Fig. 2 of the Deutsch paper).
%p A214567 with(numtheory): a := proc (n) local FS: FS := proc (n) options operator, arrow: factorset(n) end proc: if n = 1 then 1 elif bigomega(n) = 1 then 1+a(pi(n)) else 1+add(a(FS(n)[j])-1, j = 1 .. nops(FS(n))) end if end proc: seq(a(n), n = 1 .. 130);
%t A214567 a[n_] := Which[n == 1, 1, PrimeQ[n], 1 + a[PrimePi[n]], True, 1 + Total[a[#] - 1& /@ FactorInteger[n][[All, 1]]]];
%t A214567 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jun 25 2024 *)
%o A214567 (Haskell)
%o A214567 import Data.List (genericIndex)
%o A214567 a214567 n = genericIndex a214567_list (n - 1)
%o A214567 a214567_list = 1 : g 2 where
%o A214567   g x = y : g (x + 1) where
%o A214567     y | t > 0     = a214567 t + 1
%o A214567       | otherwise = 1 + sum (map ((subtract 1) . a214567) $ a027748_row x)
%o A214567        where t = a049084 x
%o A214567 -- _Reinhard Zumkeller_, Sep 03 2013
%o A214567 (PARI) a(n) = 1 + vecsum([self()(primepi(p)) |p<-factor(n)[,1]]); \\ _Kevin Ryde_, Oct 19 2022
%Y A214567 Cf. A000107.
%Y A214567 Cf. A049084, A027748.
%K A214567 nonn
%O A214567 1,2
%A A214567 _Emeric Deutsch_, Jul 25 2012