A214572 The Matula-Goebel numbers of the rooted trees having 8 vertices.

45, 50, 54, 55, 60, 63, 65, 66, 69, 70, 72, 77, 78, 80, 84, 85, 87, 88, 91, 92, 93, 94, 95, 96, 97, 98, 102, 103, 104, 111, 112, 113, 114, 116, 119, 122, 123, 124, 128, 129, 133, 136, 137, 142, 146, 148, 149, 151, 152, 158, 159, 164, 166, 167, 172, 173, 177, 178, 181, 193, 199, 201, 202, 211, 212, 214, 218, 223, 227, 233, 236, 239, 254, 262, 263, 268, 269, 271, 278, 283, 293, 311, 314, 326, 337, 353, 358, 367, 373, 382, 383, 401, 421, 431, 443, 461, 482, 547, 554, 577, 587, 599, 647, 662, 709, 739, 757, 797, 919, 967, 1063, 1153, 1523, 1787, 2221

Offset: 1

Author

Emeric Deutsch, Aug 14 2012

Keywords

• nonn
• fini
• full

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.

It is a finite sequence; number of entries is 115 = A000081(8).

Examples

			128=2^7 is in the sequence; it is the Matula-Goebel number of the star K_{1,7}.
		

Links

• E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288 [math.CO], 2011.
• E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
• 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 Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A005517, A005518, A061775, A000081.

Row n=8 of A061773. - Alois P. Heinz, Sep 06 2012

Programs

Maple

with(numtheory): N := 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+N(pi(n)) else N(r(n))+N(s(n))-1 end if end proc: A := {}: for n to 3000 do if N(n) = 8 then A := `union`(A, {n}) else  end if end do: A;

Mathematica

MGweight[n_] := If[n == 1, 1, 1 + Total[Cases[FactorInteger[n], {p_, k_} :> k*MGweight[PrimePi[p]]]]];
Select[Range[Nest[Prime, 8, 4]], MGweight[#] == 8&] (* Jean-François Alcover, Nov 11 2017, after Gus Wiseman's program for A061773 *)

Formula

A061775(n) yields the number of vertices of the rooted tree with Matula-Goebel number n. We use it to find the Matula-Goebel numbers of the rooted trees having 8 vertices.