login
Arithmetic derivative of n!.
8

%I #34 Feb 06 2024 07:56:22

%S 0,0,1,5,44,244,2064,15168,181824,1878336,21323520,238187520,

%T 3496919040,45938949120,699188474880,11185253452800,220809635020800,

%U 3774686585241600,75413794524364800,1439264469668659200,31704771803185152000,690129227948654592000

%N Arithmetic derivative of n!.

%D Giorgio Balzarotti and Paolo P. Lava, La Derivata Arithmetica, Hoepli, Milan, p. 40.

%D Ivars Peterson, Deriving the Structure of Numbers, Science News, March 20, 2004.

%H Alois P. Heinz, <a href="/A068311/b068311.txt">Table of n, a(n) for n = 0..450</a> (first 101 terms from T. D. Noe)

%H Linda Westrick, <a href="http://web.archive.org/web/20050426071741/http://web.mit.edu:80/lwest/www/intmain.pdf">Investigations of the Number Derivative</a>

%F a(n) = A003415(A000142(n)).

%e a(4) = d(4!) = d(3!*4) = d(3!)*4 + 3!*d(4) =

%e = d(2!*3)*4 + 3!*d(2*2) = d(2*3)*4 + 6*d(2*2) =

%e = (d(2)*3 + 2*d(3))*4 + 6*(d(2)*2 + 2*(d(2)) =

%e = (1*3 + 2*1)*4 + 6*(2*2*1) = 5*4 + 6*4 = 44;

%e where d(n) = A003415(n) with d(1)=0, d(prime)=1 and d(m*n)= d (m)*n + m*d(n).

%e a(6)=2064 because the arithmetic derivative of 6!=720 is 720*(4/2 + 2/3 + 1/5).

%p d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):

%p a:= proc(n) option remember;

%p `if`(n<2, 0, a(n-1)*n+(n-1)!*d(n))

%p end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jun 06 2015

%t a[0] = 0; a[1] = 0; a[n_] := Module[{f = Transpose[ FactorInteger[n]]}, If[PrimeQ[n], 1, Plus @@ (n*f[[2]]/f[[1]])]]; Table[ a[n! ], {n, 0, 6}] (* _Robert G. Wilson v_, Nov 11 2004 *)

%o (Magma) Ad:=func<h | h*(&+[Factorisation(h)[i][2]/Factorisation(h)[i][1]: i in [1..#Factorisation(h)]])>; [n le 1 select 0 else Ad(Factorial(n)): n in [0..30]]; // _Bruno Berselli_, Oct 23 2013

%o (Python 3.8+)

%o from collections import Counter

%o from math import factorial

%o from sympy import factorint

%o def A068311(n): return sum((factorial(n)*e//p for p,e in sum((Counter(factorint(m)) for m in range(2,n+1)),start=Counter({2:0})).items())) if n > 1 else 0 # _Chai Wah Wu_, Jun 12 2022

%Y Cf. A000142, A003415.

%K nonn

%O 0,4

%A _Reinhard Zumkeller_, Feb 25 2002

%E a(19)-a(21) from _Bruno Berselli_, Oct 23 2013