cp's OEIS Frontend

A068311 Arithmetic derivative of n!.

%I A068311 #42 Jan 09 2025 09:16:33
%S A068311 0,0,1,5,44,244,2064,15168,181824,1878336,21323520,238187520,
%T A068311 3496919040,45938949120,699188474880,11185253452800,220809635020800,
%U A068311 3774686585241600,75413794524364800,1439264469668659200,31704771803185152000,690129227948654592000
%N A068311 Arithmetic derivative of n!.
%D A068311 Giorgio Balzarotti and Paolo P. Lava, La Derivata Arithmetica, Hoepli, Milan, p. 40.
%D A068311 Ivars Peterson, Deriving the Structure of Numbers, Science News, March 20, 2004.
%H A068311 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 A068311 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 A068311 a(n) = A003415(A000142(n)).
%e A068311 a(4) = d(4!) = d(3!*4) = d(3!)*4 + 3!*d(4) =
%e A068311   = d(2!*3)*4 + 3!*d(2*2) = d(2*3)*4 + 6*d(2*2) =
%e A068311   = (d(2)*3 + 2*d(3))*4 + 6*(d(2)*2 + 2*d(2)) =
%e A068311   = (1*3 + 2*1)*4 + 6*(2*2*1) = 5*4 + 6*4 = 44;
%e A068311 where d(n) = A003415(n) with d(1)=0, d(prime)=1 and d(m*n)= d (m)*n + m*d(n).
%e A068311 a(6)=2064 because the arithmetic derivative of 6!=720 is 720*(4/2 + 2/3 + 1/5).
%p A068311 d:= n-> n*add(i[2]/i[1], i=ifactors(n)[2]):
%p A068311 a:= proc(n) option remember;
%p A068311       `if`(n<2, 0, a(n-1)*n+(n-1)!*d(n))
%p A068311     end:
%p A068311 seq(a(n), n=0..25);  # _Alois P. Heinz_, Jun 06 2015
%t A068311 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 A068311 (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 A068311 (Python)
%o A068311 from collections import Counter
%o A068311 from math import factorial
%o A068311 from sympy import factorint
%o A068311 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 A068311 Cf. A000142, A003415.
%K A068311 nonn
%O A068311 0,4
%A A068311 _Reinhard Zumkeller_, Feb 25 2002
%E A068311 a(19)-a(21) from _Bruno Berselli_, Oct 23 2013