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%I #16 Mar 07 2017 06:19:38
%S 0,1,1,1,1,2,1,1,2,2,1,2,2,2,2,1,1,3,1,2,2,2,2,2,2,3,3,2,2,3,1,1,2,2,
%T 2,3,2,2,3,2,2,3,2,2,3,3,2,2,2,3,2,3,1,4,2,2,2,3,1,3,3,2,3,1,3,3,1,2,
%U 3,3,2,3,2,3,3,2,2,4,2,2,4,3,2,3,2,3,3,2,2,4,3,3,2,3,2,2,2,3,3,3,3,3,3,3,3,2,2,4,2,3
%N Number of leaf-parents of the rooted tree with Matula-Goebel number n.
%C A leaf-parent in a rooted tree is a node that is the parent of at least one leaf.
%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.
%D F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
%D I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
%D I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
%D D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
%H Reinhard Zumkeller, <a href="/A196062/b196062.txt">Table of n, a(n) for n = 1..10000</a>
%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288, 2011
%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%F a(1)=0; a(2)=1; if n=p(t) (the t-th prime, t>1), then a(n)=a(t); if n=rs (r,s >=2) and both r and s are even, then a(n)=a(r)+a(s)-1; if n=rs (r,s >=2) and not both r and s are even, then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.
%e a(7)=1 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
%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 n = 2 then 1 elif bigomega(n) = 1 then a(pi(n)) elif `mod`(r(n), 2) = 0 and `mod`(s(n), 2) = 0 then a(r(n))+a(s(n))-1 else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);
%o (Haskell)
%o import Data.List (genericIndex)
%o a196062 n = genericIndex a196062_list (n - 1)
%o a196062_list = 0 : 1 : g 3 where
%o g x = y : g (x + 1) where
%o y | t > 0 = a196062 t
%o | otherwise = a196062 r + a196062 s - 0 ^ (x `mod` 4)
%o where t = a049084 x; r = a020639 x; s = x `div` r
%o -- _Reinhard Zumkeller_, Sep 03 2013
%Y Cf. A049084, A020639.
%K nonn
%O 1,6
%A _Emeric Deutsch_, Sep 30 2011
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