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 A196065 The 1st multiplicative Zagreb index of the rooted tree with Matula-Goebel number n. 2
 0, 1, 4, 4, 16, 16, 9, 9, 64, 64, 64, 36, 36, 36, 256, 16, 36, 144, 16, 144, 144, 256, 144, 64, 1024, 144, 576, 81, 144, 576, 256, 25, 1024, 144, 576, 256, 64, 64, 576, 256, 144, 324, 81, 576, 2304, 576, 576, 100, 324, 2304, 576, 324, 25, 1024, 4096, 144, 256, 576, 144, 1024, 256, 1024, 1296, 36, 2304 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS The 1st multiplicative Zagreb index of a connected graph is the product of the squared degrees of the vertices of the graph. Alternatively, it is the square of the Narumi-Katayama index. 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. LINKS E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011. F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143. I. Gutman, Multiplicative Zagreb indices of trees, Bulletin of International Mathematical Virtual Institut ISSN 1840-4367, Vol. 1, 2011, 13-19. 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. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273. H. Narumi and M. Katayama, Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Engin. Hokkaido Univ., 16, 1984, 209-214. Z. Tomovic and I. Gutman, Narumi-Katayama index of phenylenes, J. Serb. Chem. Soc., 66(4), 2001, 243-247. FORMULA a(1)=0; a(2)=1, if n = prime(t) (the t-th prime, t>=2), then a(n)=a(t)*(1+G(t))^2/G(t)^2; if n=rs (r,s>=2), then a(n)=a(r)*a(s)*G(n)^2/[G(r)*G(s)]^2; G(m) denotes the number of prime divisors of m counted with multiplicities. The Maple program is based on this recursive formula. a(n) = (A196063(n))^2. EXAMPLE a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*9*1*1=9). a(2^m) = m^2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges. MAPLE 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)))^2/bigomega(pi(n))^2 else a(r(n))*a(s(n))*bigomega(n)^2/(bigomega(r(n))^2*bigomega(s(n))^2) end if end proc: seq(a(n), n = 1 .. 65); CROSSREFS Cf. A196063, A196064. Sequence in context: A333169 A075882 A125757 * A258722 A264039 A196064 Adjacent sequences:  A196062 A196063 A196064 * A196066 A196067 A196068 KEYWORD nonn AUTHOR Emeric Deutsch, Oct 01 2011 STATUS approved

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Last modified June 2 14:21 EDT 2020. Contains 334787 sequences. (Running on oeis4.)