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%I #25 Apr 16 2019 11:22:56
%S 0,1,2,2,4,4,3,3,8,8,8,6,6,6,16,4,6,12,4,12,12,16,12,8,32,12,24,9,12,
%T 24,16,5,32,12,24,16,8,8,24,16,12,18,9,24,48,24,24,10,18,48,24,18,5,
%U 32,64,12,16,24,12,32,16,32,36,6,48,48,8,18,48,36,16,20,18,16,96,12,48,36,24,20,64,24,24,24,48,18,48,32,10,64
%N The Narumi-Katayama index of the rooted tree with Matula-Goebel number n.
%C The Narumi-Katayama index of a connected graph is the product of the degrees of the vertices of the graph.
%C 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.
%H Reinhard Zumkeller, <a href="/A196063/b196063.txt">Table of n, a(n) for n = 1..10000</a>
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H E. Deutsch, <a href="https://doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.
%H 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 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 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 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 H. Narumi and M. Katayama, <a href="http://hdl.handle.net/2115/38010">Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons</a>, Mem. Fac. Engin. Hokkaido Univ., 16, 1984, 209-214.
%H Z. Tomovic and I. Gutman, <a href="https://www.shd.org.rs/JSCS/Vol66/No4.htm#Narumi">Narumi-Katayama index of phenylenes</a>, J. Serb. Chem. Soc., 66(4), 2001, 243-247.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(1)=0; a(2)=1; if n = prime(t) (the t-th prime, t>=2), then a(n)=a(t)*(1+G(t))/G(t); if n=rs (r,s>=2), then a(n)=a(r)*a(s)*G(n)/[G(r)*G(s)]; G(m) denotes the number of prime divisors of m counted with multiplicities. The Maple program is based on this recursive formula.
%e a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*3*1*1=3).
%e a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%p 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 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n))*(1+bigomega(pi(n)))/bigomega(pi(n)) else a(r(n))*a(s(n))*bigomega(n)/(bigomega(r(n))*bigomega(s(n))) end if end proc: seq(a(n), n = 1 .. 90);
%o (Haskell)
%o import Data.List (genericIndex)
%o a196063 n = genericIndex a196063_list (n - 1)
%o a196063_list = 0 : 1 : g 3 where
%o g x = y : g (x + 1) where
%o y | t > 0 = a196063 t * (a001222 t + 1) `div` a001222 t
%o | otherwise = a196063 r * a196063 s * a001222 x `div`
%o (a001222 r * a001222 s)
%o where t = a049084 x; r = a020639 x; s = x `div` r
%o -- _Reinhard Zumkeller_, Sep 03 2013
%Y Cf. A049084, A020639, A001222, A196065.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Oct 01 2011
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