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%I #30 Aug 01 2018 18:14:04
%S 1,2,3,4,4,6,5,8,9,8,5,12,7,10,12,16,6,18,9,16,15,10,10,24,16,14,27,
%T 20,9,24,6,32,15,12,20,36,13,18,21,32,8,30,11,20,36,20,13,48,25,32,18,
%U 28,17,54,20,40,27,18,7,48,19,12,45,64,28,30,10,24,30,40,17,72,16,26,48,36,25,42,11,64,81,16,11,60,24
%N Number of root subtrees in the rooted tree with Matula-Goebel number n.
%C A root subtree of a rooted tree T is a subtree of T containing the root.
%C Also number of antichains in the rooted tree with Matula-Goebel number n. (The vertices of a rooted tree can be regarded as a partially ordered set, where u<=v holds for two vertices u and v if and only if u lies on the unique path between v and the root; an antichain is a nonempty set of mutually incomparable vertices). Example: a(5)=4 because the rooted tree with Matula-Goebel number 5 is a path ABCD and the only antichains are the 1-element subsets of {A,B,C,D}.
%C There is a simple bijection between the root subtrees and the antichains of a rooted tree: the leaves of a root subtree form an antichain (supplied by Stephan Wagner).
%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 Alois P. Heinz, <a href="/A184160/b184160.txt">Table of n, a(n) for n = 1..10000</a>
%H É. Czabarka, L. Székely, and S. Wagner, <a href="http://dx.doi.org/10.1016/j.dam.2009.07.004">The inverse problem for certain tree parameters</a>, Discrete Appl. Math., 157, 2009, 3314-3319.
%H E. Deutsch, <a href="https://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%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 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(1)=1; if n=p(t) (=the t-th prime), then a(n)=1+a(t); if n=rs (r,s,>=2), then a(n)=a(r)a(s). The Maple program is based on this recursive formula.
%F Completely multiplicative with a(prime(t)) = 1 + a(t). - _Andrew Howroyd_, Aug 01 2018
%e a(2^m) = 2^m because the rooted tree with Matula-Goebel number 2^m is a star with m edges (each edge can be included or not in the subtree).
%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 1 elif bigomega(n) = 1 then 1+a(pi(n)) else a(r(n))*a(s(n)) end if end proc: seq(a(n), n = 1 .. 85);
%t a[1] = 1; a[p_?PrimeQ] := a[p] = 1+a[PrimePi[p]]; a[n_] := a[n] = With[{f = FactorInteger[n]}, Times @@ ((a /@ f[[All, 1]])^f[[All, 2]])]; Array[a, 100] (* _Jean-François Alcover_, May 03 2017 *)
%o (PARI) a(n)={my(f=factor(n)); prod(i=1, #f~, my([p,e]=f[i,]); (1 + a(primepi(p)))^e)} \\ _Andrew Howroyd_, Aug 01 2018
%Y Cf. A184161.
%K nonn,mult
%O 1,2
%A _Emeric Deutsch_, Oct 19 2011
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