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A196053 The first Zagreb index of the rooted tree with Matula-Goebel number n.

%I A196053 #26 Jun 26 2024 02:24:38
%S A196053 0,2,6,6,10,10,12,12,14,14,14,16,16,16,18,20,16,20,20,20,20,18,20,24,
%T A196053 22,20,24,22,20,24,18,30,22,20,24,28,24,24,24,28,20,26,22,24,28,24,24,
%U A196053 34,26,28,24,26,30,32,26,30,28,24,20,32,28,22,30,42,28,28,24,26,28,30,28,38,26,28,32,30,28,30,24,38,36,24,24,34,28,26,28,32,34,36,30,30,26,28,32,46,28,32,32,36
%N A196053 The first Zagreb index of the rooted tree with Matula-Goebel number n.
%C A196053 The first Zagreb index of a simple connected graph is the sum of the squared degrees of its vertices.
%C A196053 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 A196053 Reinhard Zumkeller, <a href="/A196053/b196053.txt">Table of n, a(n) for n = 1..10000</a>
%H A196053 Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%H A196053 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 A196053 Ivan Gutman and Kinkar C. Das, <a href="http://match.pmf.kg.ac.rs/electronic_versions/Match50/match50_83-92.pdf">The first Zagreb index 30 years after</a>, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
%H A196053 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 A196053 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 A196053 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 A196053 S. Nikolic, G. Kovacevic, A. Milicevic, and N. Trinajstic, <a href="http://fulir.irb.hr/751/1/CCA_76_2003_113_124_nikolic.pdf">The Zagreb indices 30 years after</a>, Croatica Chemica Acta, 76, 2003, 113-124.
%H A196053 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A196053 a(1)=0; if n = prime(t) (the t-th prime), then a(n)=a(t)+2+2G(t); if n=r*s (r,s>=2), then a(n)=a(r)+a(s)-G(r)^2-G(s)^2+G(n)^2;  G(m) is the number of prime factors of m, counted with multiplicities. The Maple program is based on this recursive formula.
%e A196053 a(7)=12 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1+9+1+1=12).
%e A196053 a(2^m) = m(m+1) because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%p A196053 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 bigomega(n) = 1 then a(pi(n))+2+2*bigomega(pi(n)) else a(r(n))+a(s(n))-bigomega(r(n))^2-bigomega(s(n))^2+bigomega(n)^2 end if end proc: seq(a(n), n = 1 .. 100);
%t A196053 r[n_] := FactorInteger[n][[1, 1]];
%t A196053 s[n_] := n/r[n];
%t A196053 a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, a[PrimePi[n]] + 2 + 2*PrimeOmega[PrimePi[n]], True, a[r[n]] + a[s[n]] - PrimeOmega[r[n]]^2 - PrimeOmega[s[n]]^2 + PrimeOmega[n]^2];
%t A196053 Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Jun 25 2024, after Maple code *)
%o A196053 (Haskell)
%o A196053 import Data.List (genericIndex)
%o A196053 a196053 n = genericIndex a196053_list (n - 1)
%o A196053 a196053_list = 0 : g 2 where
%o A196053    g x = y : g (x + 1) where
%o A196053      y | t > 0     = a196053 t + 2 + 2 * a001222 t
%o A196053        | otherwise = a196053 r + a196053 s -
%o A196053                      a001222 r ^ 2  - a001222 s ^ 2 + a001222 x ^ 2
%o A196053        where t = a049084 x; r = a020639 x; s = x `div` r
%o A196053 -- _Reinhard Zumkeller_, Sep 03 2013
%Y A196053 Cf. A196054.
%Y A196053 Cf. A049084, A020639, A001222.
%K A196053 nonn
%O A196053 1,2
%A A196053 _Emeric Deutsch_, Sep 28 2011