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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 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=0..57.

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}]

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, A325543.

Sequence in context: A162800 A083652 A296349 * A118103 A337134 A185601

Adjacent sequences:  A325541 A325542 A325543 * A325545 A325546 A325547

KEYWORD

nonn

AUTHOR

Gus Wiseman, May 09 2019

STATUS

approved

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Last modified September 21 01:46 EDT 2021. Contains 347596 sequences. (Running on oeis4.)