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

%I A214572 #23 Nov 11 2017 11:50:20
%S A214572 45,50,54,55,60,63,65,66,69,70,72,77,78,80,84,85,87,88,91,92,93,94,95,
%T A214572 96,97,98,102,103,104,111,112,113,114,116,119,122,123,124,128,129,133,
%U A214572 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
%N A214572 The Matula-Goebel numbers of the rooted trees having 8 vertices.
%C A214572 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 A214572 It is a finite sequence; number of entries is 115 = A000081(8).
%H A214572 E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Rooted tree statistics from Matula numbers</a>, arXiv:1111.4288 [math.CO], 2011.
%H A214572 E. Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.
%H A214572 F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.
%H A214572 I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.
%H A214572 I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.
%H A214572 D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.
%H A214572 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F A214572 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.
%e A214572 128=2^7 is in the sequence; it is the Matula-Goebel number of the star K_{1,7}.
%p A214572 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;
%t A214572 MGweight[n_] := If[n == 1, 1, 1 + Total[Cases[FactorInteger[n], {p_, k_} :> k*MGweight[PrimePi[p]]]]];
%t A214572 Select[Range[Nest[Prime, 8, 4]], MGweight[#] == 8&] (* _Jean-François Alcover_, Nov 11 2017, after _Gus Wiseman_'s program for A061773 *)
%Y A214572 Cf. A005517, A005518, A061775, A000081.
%Y A214572 Row n=8 of A061773. - _Alois P. Heinz_, Sep 06 2012
%K A214572 nonn,fini,full
%O A214572 1,1
%A A214572 _Emeric Deutsch_, Aug 14 2012