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A325543
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Width (number of leaves) of the rooted tree with Matula-Goebel number n!.
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5
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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, 55, 58, 61, 65, 67, 70, 71, 76, 78, 81, 84, 88, 91, 95, 98, 102, 104, 108, 111, 114, 117, 120, 122, 127, 131, 134, 137, 141, 145, 149, 151, 156, 160, 163, 165, 169, 172
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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FORMULA
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For n > 1, a(n) = - 1 + Sum_{k = 1..n} A109129(k).
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EXAMPLE
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Matula-Goebel trees of the first 9 factorial numbers 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 leaves is the number of o's, which are (1, 1, 1, 2, 4, 5, 7, 9, 12, ...), as required.
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MATHEMATICA
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mglv[n_]:=If[n==1, 1, Total[Cases[FactorInteger[n], {p_, k_}:>mglv[PrimePi[p]]*k]]];
Table[mglv[n!], {n, 0, 100}]
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CROSSREFS
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Factorial numbers: A000142, A011371, A022559, A071626, A076934, A115627, A325272, A325273, A325276, A325508, A325544.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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