OFFSET
1,2
COMMENTS
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.
The sequence is infinite.
REFERENCES
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.
FORMULA
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.
EXAMPLE
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.
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.
From Gus Wiseman, Mar 30 2018: (Start)
Sequence of trees begins:
01 o
02 (o)
03 ((o))
04 (oo)
05 (((o)))
07 ((oo))
08 (ooo)
09 ((o)(o))
11 ((((o))))
16 (oooo)
17 (((oo)))
19 ((ooo))
21 ((o)(oo))
23 (((o)(o)))
25 (((o))((o)))
27 ((o)(o)(o))
31 (((((o)))))
(End)
MAPLE
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;
MATHEMATICA
primeMS[n_]:=If[n===1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
dep[n_]:=If[n===1, 0, 1+Max@@dep/@primeMS[n]];
rnkQ[n_]:=And[SameQ@@dep/@primeMS[n], And@@rnkQ/@primeMS[n]];
Select[Range[2000], rnkQ] (* Gus Wiseman, Mar 30 2018 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Oct 07 2011
STATUS
approved