A068311 Arithmetic derivative of n!.

0, 0, 1, 5, 44, 244, 2064, 15168, 181824, 1878336, 21323520, 238187520, 3496919040, 45938949120, 699188474880, 11185253452800, 220809635020800, 3774686585241600, 75413794524364800, 1439264469668659200, 31704771803185152000, 690129227948654592000

Offset: 0

Author

Reinhard Zumkeller, Feb 25 2002

Keywords

• nonn

Examples

			a(4) = d(4!) = d(3!*4) = d(3!)*4 + 3!*d(4) =
  = d(2!*3)*4 + 3!*d(2*2) = d(2*3)*4 + 6*d(2*2) =
  = (d(2)*3 + 2*d(3))*4 + 6*(d(2)*2 + 2*d(2)) =
  = (1*3 + 2*1)*4 + 6*(2*2*1) = 5*4 + 6*4 = 44;
where d(n) = A003415(n) with d(1)=0, d(prime)=1 and d(m*n)= d (m)*n + m*d(n).
a(6)=2064 because the arithmetic derivative of 6!=720 is 720*(4/2 + 2/3 + 1/5).
		

References

• Giorgio Balzarotti and Paolo P. Lava, La Derivata Arithmetica, Hoepli, Milan, p. 40.
• Ivars Peterson, Deriving the Structure of Numbers, Science News, March 20, 2004.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
• Linda Westrick, Investigations of the Number Derivative

Crossrefs

Cf. A000142, A003415.

Programs

Magma

Ad:=func; [n le 1 select 0 else Ad(Factorial(n)): n in [0..30]]; // Bruno Berselli, Oct 23 2013

Maple

d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
a:= proc(n) option remember;
      `if`(n<2, 0, a(n-1)*n+(n-1)!*d(n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Jun 06 2015

Mathematica

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 *)

Python

from collections import Counter
from math import factorial
from sympy import factorint
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

Formula

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

Extensions

a(19)-a(21) from Bruno Berselli, Oct 23 2013