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%I #19 Mar 31 2018 17:00:27
%S 1,2,3,4,5,7,8,9,11,16,17,19,21,23,25,27,31,32,49,53,57,59,63,64,67,
%T 73,81,83,85,97,103,115,121,125,127,128,131,133,147,159,171,189,227,
%U 241,243,256,269,277,289,307,311,331,335,343,361,365,367,371,391,393,399,419,425,431,439,441,477
%N The Matula-Goebel number of rooted trees having all leaves at the same level.
%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.
%C The sequence is infinite.
%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F In A184154 one constructs for each n the generating polynomial P(n,x) of the leaves of the rooted tree with Matula-Goebel number n, according to their levels. The Maple program finds those n (between 1 and 500) for which P(n,x) is a monomial.
%e 7 is in the sequence because the rooted tree with Matula-Goebel number 7 is the rooted tree Y, having all leaves at level 2.
%e 2^m is in the sequence for each positive integer m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%e From _Gus Wiseman_, Mar 30 2018: (Start)
%e Sequence of trees begins:
%e 01 o
%e 02 (o)
%e 03 ((o))
%e 04 (oo)
%e 05 (((o)))
%e 07 ((oo))
%e 08 (ooo)
%e 09 ((o)(o))
%e 11 ((((o))))
%e 16 (oooo)
%e 17 (((oo)))
%e 19 ((ooo))
%e 21 ((o)(oo))
%e 23 (((o)(o)))
%e 25 (((o))((o)))
%e 27 ((o)(o)(o))
%e 31 (((((o)))))
%e (End)
%p with(numtheory): P := 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 sort(expand(x*P(pi(n)))) else sort(P(r(n))+P(s(n))) end if end proc: A := {}: for n to 500 do if degree(numer(subs(x = 1/x, P(n)))) = 0 then A := `union`(A, {n}) else end if end do: A;
%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t dep[n_]:=If[n===1,0,1+Max@@dep/@primeMS[n]];
%t rnkQ[n_]:=And[SameQ@@dep/@primeMS[n],And@@rnkQ/@primeMS[n]];
%t Select[Range[2000],rnkQ] (* _Gus Wiseman_, Mar 30 2018 *)
%Y Cf. A000081, A003238, A004111, A007097, A048816, A061775, A109082, A184154, A214577, A244925, A276625, A290689, A290760, A290822, A298422, A298424, A298426.
%K nonn
%O 1,2
%A _Emeric Deutsch_, Oct 07 2011
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