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

0,4

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

Formula

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

Example

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).

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 *)

Prog

(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
(Python 3.8+)
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

Crossrefs

Cf. A000142, A003415.

Sequence in context: A271298 A271118 A268762 * A109984 A247776 A374300

Adjacent sequences: A068308 A068309 A068310 * A068312 A068313 A068314

Keyword

nonn

Author

Reinhard Zumkeller, Feb 25 2002

Extensions

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

Status

approved