%I M1154 #50 Jan 24 2022 07:11:34
%S 1,2,4,8,19,67,331,2221,19577,219613,3042161,50728129,997525853,
%T 22742734291,592821132889,17461204521323,575411103069067,
%U 21034688742654437,846729487306354343
%N Largest label f(T) given to a rooted tree T with n nodes in Matula-Goebel labeling.
%C Let prime(1)=2, ... denote the primes. The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product prime(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%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, Vol. 29, No. 1 (1980), pp. 141-143.
%H I. Gutman and A. Ivić, <a href="https://www.jstor.org/stable/44095760">Graphs with maximal and minimal Matula numbers</a>, Bulletin CVII Acad. Serbe, Sciences Math., Vol. 107, No. 19 (1994), pp. 65-74.
%H I. Gutman and A. Ivić, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., Vol. 150, No. 1-3 (1996), pp. 131-142.
%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev., Vol. 10 (1968), p. 273.
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>.
%H <a href="/index/Ro#rooted">Index entries for sequences related to rooted trees</a>.
%H <a href="/index/Tra#trees">Index entries for sequences related to trees</a>.
%F a(1)=1; a(2)=2; a(3)=4; a(4)=8; a(n) = the a(n-1)-th prime (see the Gutman and Ivic 1994 paper). - _Emeric Deutsch_, Apr 15 2012
%F Under plausible assumptions about the growth of the primes, for n >= 4, a(n+1) = a(n)-th prime and A005518(n) = A057452(n-3). - _David W. Wilson_, Jul 09 2001
%F A091233(n) = (a(n)-A005517(n))+1. - _Antti Karttunen_, May 24 2004
%p with(numtheory): a := proc (n) if n = 1 then 1 elif n = 2 then 2 elif n = 3 then 4 elif n = 4 then 8 else ithprime(a(n-1)) end if end proc: seq(a(n), n = 1 .. 12); # _Emeric Deutsch_, Apr 15 2012
%t a[n_] := a[n] = Switch[n, 1, 1, 2, 2, 3, 4, 4, 8, _, Prime[a[n-1]]]; Table[a[n], {n, 1, 16}] (* _Jean-François Alcover_, Mar 06 2014, after _Emeric Deutsch_ *)
%Y Apart from initial terms, same as A057452.
%Y Cf. A061773. See A005517 for the smallest value of f(T).
%K nonn
%O 1,2
%A _N. J. A. Sloane_
%E More terms from _David W. Wilson_, Jul 09 2001
%E a(17)-a(19) from _Robert G. Wilson v_, Mar 07 2017 using Kim Walisch's primecount