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A068311
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Arithmetic derivative of n!.
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8
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0, 0, 1, 5, 44, 244, 2064, 15168, 181824, 1878336, 21323520, 238187520, 3496919040, 45938949120, 699188474880, 11185253452800, 220809635020800, 3774686585241600, 75413794524364800, 1439264469668659200, 31704771803185152000, 690129227948654592000
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OFFSET
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0,4
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REFERENCES
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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.
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LINKS
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FORMULA
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EXAMPLE
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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).
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MAPLE
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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:
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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