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Search: id:A068311
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| A068311 |
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Arithmetic derivative of n!. |
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+0
3
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| 0, 0, 1, 5, 44, 244, 2064, 15168, 181824, 1878336, 21323520, 238187520, 3496919040, 45938949120, 699188474880, 11185253452800, 220809635020800, 3774686585241600, 75413794524364800
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OFFSET
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0,4
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REFERENCES
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Ivars Peterson, Deriving the Structure of Numbers, Science News, March 20, 2004.
Linda Westrick, Investigations of the Number Derivative (pdf)
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
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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 arithmetic derivative of 6!=720 is 720(4/2+2/3+1/5).
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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}] (from Robert G. Wilson v Nov 11 2004)
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CROSSREFS
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Equals A003415(A000142(n))
Sequence in context: A071861 A159298 A128523 this_sequence A109984 A096355 A054766
Adjacent sequences: A068308 A068309 A068310 this_sequence A068312 A068313 A068314
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KEYWORD
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nonn
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AUTHOR
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Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Feb 25 2002
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