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A093101 Cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms. 22
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Same as n!/A061355(n) and (1+n+n(n-1)+n(n-1)(n-2)+...+n!)/A061354(n).
a(n) is relatively prime to n.
gcd(a(n),a(n+1)) = 1.
LINKS
J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.
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.
FORMULA
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.
EXAMPLE
E.g. 1/0!+1/1!+1/2!+1/3!=16/6=(2*8)/(2*3) so a(3)=2.
MATHEMATICA
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}]
PROG
(PARI)
A000522(n) = sum(k=0, n, binomial(n, k)*k!); \\ This function from Joerg Arndt, Dec 14 2014
A093101(n) = gcd(n!, A000522(n)); \\ Antti Karttunen, Jul 12 2017
CROSSREFS
(n+1)!/(a(n)*a(n+1)) = A123899(n).
(n+3)!/(a(n)*a(n+1)*a(n+2)) = A123900(n).
(n+3)/GCD(a(n), a(n+2)) = A123901(n).
Cf. also A000522, A061354, A061355.
Sequence in context: A287541 A288196 A072883 * A082469 A206566 A088151
KEYWORD
nonn
AUTHOR
Jonathan Sondow, May 10 2004, Oct 18 2006
EXTENSIONS
More terms from Robert G. Wilson v, May 14 2004
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)