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A196065
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The 1st multiplicative Zagreb index of the rooted tree with Matula-Goebel number n.
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2
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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
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MAPLE
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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);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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