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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 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.)