A184155 The Matula-Goebel number of rooted trees having all leaves at the same level.

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, 73, 81, 83, 85, 97, 103, 115, 121, 125, 127, 128, 131, 133, 147, 159, 171, 189, 227, 241, 243, 256, 269, 277, 289, 307, 311, 331, 335, 343, 361, 365, 367, 371, 391, 393, 399, 419, 425, 431, 439, 441, 477

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.

Links

Table of n, a(n) for n=1..67.

Index entries for sequences related to Matula-Goebel numbers

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

Cf. A000081, A003238, A004111, A007097, A048816, A061775, A109082, A184154, A214577, A244925, A276625, A290689, A290760, A290822, A298422, A298424, A298426.

Sequence in context: A316468 A099627 A324841 * A331913 A318612 A318690

Adjacent sequences: A184152 A184153 A184154 * A184156 A184157 A184158

Keyword

nonn

Author

Emeric Deutsch, Oct 07 2011

Status

approved