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%I #9 May 11 2019 18:31:10
%S 1,1,1,2,4,5,7,9,12,14,16,17,20,22,25,27,31,33,36,39,42,45,47,49,53,
%T 55,58,61,65,67,70,71,76,78,81,84,88,91,95,98,102,104,108,111,114,117,
%U 120,122,127,131,134,137,141,145,149,151,156,160,163,165,169,172
%N Width (number of leaves) of the rooted tree with Matula-Goebel number n!.
%C Also the multiplicity of q(1) in the factorization of n! into factors q(i) = prime(i)/i. For example, the factorization of 7! is q(1)^9 * q(2)^3 * q(3) * q(4), so a(7) = 9.
%F For n > 1, a(n) = - 1 + Sum_{k = 1..n} A109129(k).
%e Matula-Goebel trees of the first 9 factorial numbers are:
%e 0!: o
%e 1!: o
%e 2!: (o)
%e 3!: (o(o))
%e 4!: (ooo(o))
%e 5!: (ooo(o)((o)))
%e 6!: (oooo(o)(o)((o)))
%e 7!: (oooo(o)(o)((o))(oo))
%e 8!: (ooooooo(o)(o)((o))(oo))
%e The number of leaves is the number of o's, which are (1, 1, 1, 2, 4, 5, 7, 9, 12, ...), as required.
%t mglv[n_]:=If[n==1,1,Total[Cases[FactorInteger[n],{p_,k_}:>mglv[PrimePi[p]]*k]]];
%t Table[mglv[n!],{n,0,100}]
%Y Cf. A000081, A001222, A056239, A324922, A324923, A324924.
%Y Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
%Y Factorial numbers: A000142, A011371, A022559, A071626, A076934, A115627, A325272, A325273, A325276, A325508, A325544.
%K nonn
%O 0,4
%A _Gus Wiseman_, May 09 2019
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