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The number of rooted trees that are isomorphic as trees to the rooted tree with Matula number n (n >=1).
1

%I #31 Sep 14 2024 13:21:55

%S 1,1,2,2,2,2,2,2,3,3,3,4,4,4,3,2,4,4,2,5,5,3,4,4,4,4,3,2,5,7,3,2,4,5,

%T 6,4,4,4,7,5,4,6,2,6,6,3,7,4,3,5,6,6,2,4,4,4,5,7,5,7,4,4,5,2,8,8,4,3,

%U 6,7,5,4,6,4,6,4,7

%N The number of rooted trees that are isomorphic as trees to the rooted tree with Matula number n (n >=1).

%C a(n) is the number of entries in row n of the triangle A235121.

%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.

%H Emeric Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.

%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. Matula, <a href="http://www.jstor.org/stable/2027327?seq=30">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>

%p with(numtheory): f := proc (m) local x, p, S: S := NULL: x := factorset(m): for p in x do S := S, ithprime(m/p)*pi(p) end do: S end proc: M := proc (m) local A, B: A := {m}: do B := A: A := `union`(map(f, A), A): if B = A then return A end if end do end proc: seq(nops(M(j)), j = 1 .. 100); # _W. Edwin Clark_

%t f[m_] := Prime[m/#]*PrimePi[#]& /@ FactorInteger[m][[All, 1]];

%t M[m_] := Module[{A, B}, A = {m}; While[True, B = A; A = Union[Map[f, A] // Flatten, A] // DeleteCases[#, 0]&; If[B == A, Return[A]]]];

%t Table[M[j] // Length, {j, 1, 100}] (* _Jean-François Alcover_, Sep 14 2024, after _W. Edwin Clark_ *)

%Y Cf. A235121.

%K nonn

%O 1,3

%A _Emeric Deutsch_, Jan 19 2014