

A214572


The MatulaGoebel numbers of the rooted trees having 8 vertices.


1



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
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OFFSET

1,1


COMMENTS

The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.
It is a finite sequence; number of entries is 115 = A000081(8).


LINKS

Table of n, a(n) for n=1..115.
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, 23142322.
F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

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


EXAMPLE

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


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&] (* JeanFrançois Alcover, Nov 11 2017, after Gus Wiseman's program for A061773 *)


CROSSREFS

Cf. A005517, A005518, A061775, A000081.
Row n=8 of A061773.  Alois P. Heinz, Sep 06 2012
Sequence in context: A306103 A045566 A274368 * A295493 A184043 A295802
Adjacent sequences: A214569 A214570 A214571 * A214573 A214574 A214575


KEYWORD

nonn,fini,full


AUTHOR

Emeric Deutsch, Aug 14 2012


STATUS

approved



