%I #27 Jun 25 2024 12:32:15
%S 0,1,2,2,2,2,3,3,2,2,2,3,3,3,2,4,3,3,4,3,3,2,3,4,2,3,3,3,3,3,2,5,2,3,
%T 3,4,4,4,3,4,3,3,3,3,3,3,3,5,3,3,3,3,5,4,2,4,4,3,3,4,4,2,3,6,3,3,4,3,
%U 3,3,4,5,3,4,3,4,3,3,3,5,4,3,3,4,3,3,3,4,5,4,3,3,2,3,4,6,3,3,3,4,3,3,4,4,3,5,4,5,3,3
%N Maximum vertex-degree in the rooted tree with Matula-Goebel number n.
%C The Matula-Goebel number of a rooted tree is 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.
%H Reinhard Zumkeller, <a href="/A196046/b196046.txt">Table of n, a(n) for n = 1..10000</a>
%H Emeric Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
%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 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(1)=0; if n=prime(t) (=the t-th prime), then a(n) = max(a(t), 1+G(t)); if n=r*s (r,s>=2), then a(n)=max(a(r),a(s), G(r)+G(s)); G(m) is the number of prime divisors of m counted with multiplicity. The Maple program is based on this recursive formula.
%F The Gutman et al. references contain a different recursive formula.
%e a(7)=3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
%e a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
%p with(numtheory): a := 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 0 elif bigomega(n) = 1 then max(a(pi(n)), 1+bigomega(pi(n))) else max(a(r(n)), a(s(n)), bigomega(r(n))+bigomega(s(n))) end if end proc: seq(a(n), n = 1 .. 110);
%t r[n_] := FactorInteger[n][[1, 1]];
%t s[n_] := n/r[n];
%t a[n_] := Which[n == 1, 0, PrimeOmega[n] == 1, Max[a[PrimePi[n]], 1 + PrimeOmega[PrimePi[n]]], True, Max[a[r[n]], a[s[n]], PrimeOmega[r[n]] + PrimeOmega[s[n]]]];
%t Table[a[n], {n, 1, 110}] (* _Jean-François Alcover_, Jun 25 2024, after Maple code *)
%o (Haskell)
%o import Data.List (genericIndex)
%o a196046 n = genericIndex a196046_list (n - 1)
%o a196046_list = 0 : g 2 where
%o g x = y : g (x + 1) where
%o y | t > 0 = max (a196046 t) (a001222 t + 1)
%o | otherwise = maximum [a196046 r, a196046 s, a001222 r + a001222 s]
%o where t = a049084 x; r = a020639 x; s = x `div` r
%o -- _Reinhard Zumkeller_, Sep 03 2013
%Y Cf. A049084, A020639, A001222.
%K nonn
%O 1,3
%A _Emeric Deutsch_, Sep 26 2011