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A325544
Number of nodes in the rooted tree with Matula-Goebel number n!.
4
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, 95, 100, 106, 111, 116, 122, 127, 132, 138, 143, 149, 155, 160, 165, 171, 177, 182, 188, 193, 199, 206, 212, 218, 224, 230, 237, 243, 249, 254, 261, 268, 274, 280
OFFSET
0,3
COMMENTS
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.
FORMULA
For n > 1, a(n) = 1 - n + Sum_{k = 1..n} A061775(k).
EXAMPLE
Matula-Goebel trees of the first 9 factorial number are:
0!: o
1!: o
2!: (o)
3!: (o(o))
4!: (ooo(o))
5!: (ooo(o)((o)))
6!: (oooo(o)(o)((o)))
7!: (oooo(o)(o)((o))(oo))
8!: (ooooooo(o)(o)((o))(oo))
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.
MATHEMATICA
mgwt[n_]:=If[n==1, 1, 1+Total[Cases[FactorInteger[n], {p_, k_}:>mgwt[PrimePi[p]]*k]]];
Table[mgwt[n!], {n, 0, 100}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, May 09 2019
STATUS
approved