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A093101
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Cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.
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22
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1, 1, 1, 2, 1, 2, 1, 20, 1, 10, 1, 8, 5, 2, 5, 4, 1, 130, 1, 4000, 1, 2, 5, 52, 5, 494, 1, 40, 1, 10, 13, 4, 25, 38, 5, 16, 13, 230, 13, 20, 1, 46, 5, 104, 475, 62, 1, 20, 1, 130, 31, 832, 2755, 74, 5, 4, 13, 50, 1, 40, 23, 2, 2795, 76, 34385, 2, 1, 80, 1, 650, 1, 2812, 5, 74, 5
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OFFSET
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0,4
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COMMENTS
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a(n) is relatively prime to n.
gcd(a(n),a(n+1)) = 1.
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LINKS
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J. Sondow and K. Schalm, Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.
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FORMULA
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a(n) = gcd(n!, 1+n+n(n-1)+n(n-1)(n-2)+...+n!).
a(n) = gcd(n!, A(n)) where A(0) = 1, A(n) = n*A(n-1)+1.
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EXAMPLE
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E.g. 1/0!+1/1!+1/2!+1/3!=16/6=(2*8)/(2*3) so a(3)=2.
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MATHEMATICA
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f[n_] := n! / Denominator[ Sum[1/k!, {k, 0, n}]]; Table[ f[n], {n, 0, 74}] (* Robert G. Wilson v *)
(* Second program: *)
A[n_] := If[n==0, 1, n*A[n-1]+1]; Table[GCD[A[n], n! ], {n, 0, 74}]
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PROG
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(PARI)
A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from Joerg Arndt, Dec 14 2014
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CROSSREFS
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(n+3)!/(a(n)*a(n+1)*a(n+2)) = A123900(n).
(n+3)/GCD(a(n), a(n+2)) = A123901(n).
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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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