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A325543 Width (number of leaves) of the rooted tree with Matula-Goebel number n!. 4
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified November 21 06:00 EST 2019. Contains 329350 sequences. (Running on oeis4.)