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Number of nodes in the rooted tree with Matula-Goebel number n!.
4

%I #5 May 11 2019 18:31:19

%S 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 95,100,106,111,116,122,127,132,138,143,149,155,160,165,171,177,182,

%U 188,193,199,206,212,218,224,230,237,243,249,254,261,268,274,280

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

%C 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 For n > 1, a(n) = 1 - n + Sum_{k = 1..n} A061775(k).

%e Matula-Goebel trees of the first 9 factorial number 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 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 mgwt[n_]:=If[n==1,1,1+Total[Cases[FactorInteger[n],{p_,k_}:>mgwt[PrimePi[p]]*k]]];

%t Table[mgwt[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, A325543.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 09 2019