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A184161
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Number of subtrees in the rooted tree with Matula-Goebel number n.
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4
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1, 3, 6, 6, 10, 10, 11, 11, 15, 15, 15, 17, 17, 17, 21, 20, 17, 25, 20, 24, 24, 21, 25, 30, 28, 25, 36, 28, 24, 34, 21, 37, 28, 24, 32, 44, 30, 30, 34, 41, 25, 40, 28, 32, 48, 36, 34, 55, 37, 45, 32, 40, 37, 64, 36, 49, 41, 34, 24, 59, 44, 28, 57, 70, 44, 44, 30, 37, 48, 53, 41, 81, 40, 44, 63, 49, 41, 56, 32, 74
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OFFSET
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1,2
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COMMENTS
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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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REFERENCES
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É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 3314-3319.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
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 Review, 10, 1968, 273.
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LINKS
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Table of n, a(n) for n=1..80.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011
Index entries for sequences related to Matula-Goebel numbers
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FORMULA
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Let b(n)=A184160(n) denote the number of those subtrees of the rooted tree with Matula-Goebel number n that contain the root. Then a(1)=1; if n=p(t) (=the t=th prime), then a(n)=1+a(t)+b(t); if n=rs (r,s,>=2), then a(n)=a(r)+a(s)+(b(r)-1)(b(s)-1)-1. The Maple program is based on this recursive formula.
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EXAMPLE
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a(4) = 6 because the rooted tree with Matula-Goebel number 4 is V; it has 6 subtrees (three 1-vertex subtrees, two 1-edge subtrees, and the tree itself).
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MAPLE
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with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 1 elif bigomega(n) = 1 then 1+b(pi(n)) else b(r(n))*b(s(n)) end if end proc: if n = 1 then 1 elif bigomega(n) = 1 then a(pi(n))+b(pi(n))+1 else a(r(n))+a(s(n))+(b(r(n))-1)*(b(s(n))-1)-1 end if end proc: seq(a(n), n = 1 .. 80);
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CROSSREFS
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Cf. A184160.
Sequence in context: A108850 A111652 A159787 * A276000 A316563 A316140
Adjacent sequences: A184158 A184159 A184160 * A184162 A184163 A184164
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Oct 19 2011
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STATUS
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approved
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