|
| |
|
|
A061354
|
|
Numerator of Sum_{k=0..n} 1/k!.
|
|
17
| |
|
|
1, 2, 5, 8, 65, 163, 1957, 685, 109601, 98641, 9864101, 13563139, 260412269, 8463398743, 47395032961, 888656868019, 56874039553217, 7437374403113, 17403456103284421, 82666416490601, 6613313319248080001, 69439789852104840011
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| p divides a(p-1) for prime p = {2, 5, 13, 37, 463, ...} which apparently coincides with A064384(n) = {2, 5, 13, 37, 463, ...} Primes p such that p divides 0!-1!+2!-3!+...+(-1)^{p-1}(p-1)!. - Alexander Adamchuk (alex(AT)kolmogorov.com), Jun 14 2007
GCD(a(n), a(n+2)) = A124779(n) is either 1 or a prime 2, 5, 13, 37, 463, ... = A064384. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jun 12 2007
For proofs of Adamchuk's and my Comments, see the link "The Taylor series for e ...". - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jun 18 2007
Furdui proves that, if e_n = (1+(1/n))^n then Limit[n->infinity] n*((e_n)^e - e^(A061354(n)/A061355(n)) = -(1/2)*e^(e+1). [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 03 2008]
|
|
|
REFERENCES
| Ovidiu Furdui, Solution to problem 91.H, Mathematical Gazette 92(523), 2008, pp. 174-175. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 03 2008]
|
|
|
LINKS
| Harry J. Smith, Table of n, a(n) for n=0,...,200
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, The Taylor series for e and the primes 2, 5, 13, 37, 463, ...: a surprising connection
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.
Index entries for sequences related to factorial numbers
|
|
|
FORMULA
| a(n) = A000522(n)/A093101(n).
Numerators of floor(n!*exp(1))/n!, n>=1. Numerators of coefficients in expansion of exp(x)/(1-x). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 11 2002
(1+n+n(n-1)+...+n!)/GCD(n!,1+n+n(n-1)+...+n!) - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Aug 18 2006
|
|
|
EXAMPLE
| 1, 2, 5/2, 8/3, 65/24, 163/60, 1957/720, 685/252, ...
|
|
|
PROG
| (PARI) { default(realprecision, 500); e=exp(1); for (n=0, 200, a=numerator(floor(n!*e)/n!); if (n==0, a=1); write("b061354.txt", n, " ", a) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jul 21 2009]
|
|
|
CROSSREFS
| Cf. A061355, A093101, A064384, A064384, A124779, A129924.
Sequence in context: A191550 A120342 A180627 * A162709 A011039 A021390
Adjacent sequences: A061351 A061352 A061353 * A061355 A061356 A061357
|
|
|
KEYWORD
| nonn,frac
|
|
|
AUTHOR
| Amarnath_murthy (amarnath_murthy(AT)yahoo.com), Apr 28 2001
|
| |
|
|
