A325543 Width (number of leaves) of the rooted tree with Matula-Goebel number n!.

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

Offset

0,4

Comments

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.

Links

Table of n, a(n) for n=0..61.

Formula

For n > 1, a(n) = - 1 + Sum_{k = 1..n} A109129(k).

Example

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.

Mathematica

mglv[n_]:=If[n==1, 1, Total[Cases[FactorInteger[n], {p_, k_}:>mglv[PrimePi[p]]*k]]];
Table[mglv[n!], {n, 0, 100}]

Crossrefs

Cf. A000081, A001222, A056239, A324922, A324923, A324924.

Matula-Goebel numbers: A007097, A061775, A109082, A109129, A196050, A317713.

Factorial numbers: A000142, A011371, A022559, A071626, A076934, A115627, A325272, A325273, A325276, A325508, A325544.

Sequence in context: A000788 A053039 A286753 * A214051 A027861 A219648

Adjacent sequences: A325540 A325541 A325542 * A325544 A325545 A325546

Keyword

nonn

Author

Gus Wiseman, May 09 2019

Status

approved