

A184160


Number of root subtrees in the rooted tree with MatulaGoebel number n.


6



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, 20, 9, 24, 6, 32, 15, 12, 20, 36, 13, 18, 21, 32, 8, 30, 11, 20, 36, 20, 13, 48, 25, 32, 18, 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
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OFFSET

1,2


COMMENTS

A root subtree of a rooted tree T is a subtree of T containing the root.
Also number of antichains in the rooted tree with MatulaGoebel 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 MatulaGoebel number 5 is a path ABCD and the only antichains are the 1element subsets of {A,B,C,D}.
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).
The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..10000
É. Czabarka, L. Székely, and S. Wagner, The inverse problem for certain tree parameters, Discrete Appl. Math., 157, 2009, 33143319.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

a(1)=1; if n=p(t) (=the tth 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.
Completely multiplicative with a(prime(t)) = 1 + a(t).  Andrew Howroyd, Aug 01 2018


EXAMPLE

a(2^m) = 2^m because the rooted tree with MatulaGoebel number 2^m is a star with m edges (each edge can be included or not in the subtree).


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 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);


MATHEMATICA

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] (* JeanFrançois Alcover, May 03 2017 *)


PROG

(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


CROSSREFS

Cf. A184161.
Sequence in context: A146288 A244361 A199424 * A064553 A126012 A283267
Adjacent sequences: A184157 A184158 A184159 * A184161 A184162 A184163


KEYWORD

nonn,mult


AUTHOR

Emeric Deutsch, Oct 19 2011


STATUS

approved



