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A325544 Number of nodes in the rooted tree with Matula-Goebel number n!.

%I A325544 #5 May 11 2019 18:31:19
%S A325544 1,1,2,4,6,9,12,15,18,22,26,30,34,38,42,47,51,55,60,64,69,74,79,84,89,
%T A325544 95,100,106,111,116,122,127,132,138,143,149,155,160,165,171,177,182,
%U A325544 188,193,199,206,212,218,224,230,237,243,249,254,261,268,274,280
%N A325544 Number of nodes in the rooted tree with Matula-Goebel number n!.
%C A325544 Also one plus the number of factors in the factorization of n! into factors q(i) = prime(i)/i. For example, the q-factorization of 7! is 7! = q(1)^9 * q(2)^3 * q(3) * q(4), with 14 = a(7) - 1 factors.
%F A325544 For n > 1, a(n) = 1 - n + Sum_{k = 1..n} A061775(k).
%e A325544 Matula-Goebel trees of the first 9 factorial number are:
%e A325544   0!: o
%e A325544   1!: o
%e A325544   2!: (o)
%e A325544   3!: (o(o))
%e A325544   4!: (ooo(o))
%e A325544   5!: (ooo(o)((o)))
%e A325544   6!: (oooo(o)(o)((o)))
%e A325544   7!: (oooo(o)(o)((o))(oo))
%e A325544   8!: (ooooooo(o)(o)((o))(oo))
%e A325544 The number of nodes is the number of o's plus the number of brackets, giving {1,1,2,4,6,9,12,15,18}, as required.
%t A325544 mgwt[n_]:=If[n==1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>mgwt[PrimePi[p]]*k]]];
%t A325544 Table[mgwt[n!],{n,0,100}]
%Y A325544 Cf. A000081, A001222, A056239, A324922, A324923, A324924.
%Y A325544 Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.
%Y A325544 Factorial numbers: A000142, A011371, A022559, A071626, A076934, A115627, A325272, A325273, A325276, A325508, A325543.
%K A325544 nonn
%O A325544 0,3
%A A325544 _Gus Wiseman_, May 09 2019