%N a(n) = the largest of the M-indices of the trees with n vertices.
%C We define the M-index of a tree T to be the smallest of the Matula numbers of the rooted trees isomorphic (as a tree) to T. Example. The path tree P = ABCDE has M-index 9. Indeed, there are 3 rooted trees isomorphic to P: rooted at A, B, and C, respectively. Their Matula numbers are 11, 10, and 9, respectively. Consequently, the M-index of P is 9.
%C a(n) = largest (= last) entry in row n of A235111.
%C It is conjectured that for n>=7 one has a(n) = A235120(n-6).
%C These numbers can be useful, for example, in the following situation. We intend to identify all trees that have 10 vertices and satisfy a certain property. Instead of scanning all rooted trees with Matula numbers from A005517(10)=125 to A005518(10)=219613, we do the scanning only for Matula numbers between 125 and a(10)=1273.
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288, 2011.
%H 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 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 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 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 I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, <a href="http://dx.doi.org/10.1021/ci990060s">The multiplicative version of the Wiener index</a>, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.
%H I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, <a href="http://dx.doi.org/10.1007/PL00010312">On the multiplicative Wiener index and its possible chemical applications</a>, Monatshefte f. Chemie, 131, 2000, 421-427.
%H 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 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(n) = A235111(n,A000055(n)).
%e a(4)=7. Indeed, there are 2 trees with 4 vertices: the path P and the star S with 3 edges. There are two rooted trees isomorphic to P; they have Matula numbers 5 and 6; so the M-index is 5. There are two rooted trees isomorphic to S; they have Matula numbers 7 and 8; so the M-index is 7. Max(5,7) = 7.
%Y Cf. A235111, A005518.
%A _Emeric Deutsch_, Jan 03 2014
%E a(13) from _Emeric Deutsch_, Feb 16 2014
%E a(14) from _Emeric Deutsch_, Mar 12 2014