A007677 Denominators of convergents to e.
1, 1, 3, 4, 7, 32, 39, 71, 465, 536, 1001, 8544, 9545, 18089, 190435, 208524, 398959, 4996032, 5394991, 10391023, 150869313, 161260336, 312129649, 5155334720, 5467464369, 10622799089, 196677847971, 207300647060, 403978495031, 8286870547680, 8690849042711
Offset: 0
Examples
2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, 2721/1001, 23225/8544, 25946/9545, 49171/18089, 517656/190435, 566827/208524, 1084483/398959, 13580623/4996032, 14665106/5394991, 28245729/10391023, 410105312/150869313, 438351041/161260336, 848456353/312129649, ...
References
- CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
- W. J. LeVeque, Fundamentals of Number Theory. Addison-Wesley, Reading, MA, 1977, p. 240.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Eric W. Weisstein, Table of n, a(n) for n = 0..1000 (first 200 terms from T. D. Noe)
- L. Bayon, P. Fortuny, J. M. Grau, M. M. Ruiz, M. A. Oller-Marcen, The Best-or-Worst and the Postdoc problems with random number of candidates, arXiv:1809.06390 [math.PR], 2018.
- E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
- C. Elsner, Series of Error Terms for Rational Approximations of Irrational Numbers, J. Int. Seq. 14 (2011) # 11.1.4.
- C. Elsner, M. Stein, On Error Sum Functions Formed by Convergents of Real Numbers, J. Int. Seq. 14 (2011) # 11.8.6
- J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641 (article), 114 (2007) 659 (addendum).
- J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
- 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; arXiv:0709.0671 [math.NT], 2007-2009.
- Eric Weisstein's World of Mathematics, e Continued Fraction
- Eric Weisstein's World of Mathematics, Sultan's Dowry Problem
Programs
-
Maple
Digits := 60: E := exp(1); convert(evalf(E),confrac,50,'cvgts'): cvgts;
-
Mathematica
Denominator[Convergents[E, 40]] (* T. D. Noe, Oct 12 2011 *) Denominator[Table[Piecewise[{ {Hypergeometric1F1[-1 - n/3, -1 - (2 n)/3, 1]/Hypergeometric1F1[-(n/3), -1 - (2 n)/3, -1], Mod[n, 3] == 0}, {Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), 1]/Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), -1], Mod[n, 3] == 1}, {Hypergeometric1F1[1/3 (-1 - n), 1 - (2 (4 + n))/3, 1]/Hypergeometric1F1[1/3 (-4 - n), 1 - (2 (4 + n))/3, -1], Mod[n, 3] == 2} }], {n, 0, 30}]] (* Eric W. Weisstein, Sep 09 2013 *) Table[Piecewise[{ {(1 + (2 n)/3)!/(n/3)! Hypergeometric1F1[-(n/3), -1 - (2 n)/3, -1], Mod[n, 3] == 0}, {((2 (2 + n))/3)!/((2 + n)/3)! Hypergeometric1F1[1/3 (-2 - n), -(2/3) (2 + n), -1], Mod[n, 3] == 1}, {((4 + n) (5/3 + (2 n)/3)! )/(3 ((4 + n)/3)!) Hypergeometric1F1[1/3 (-4 - n), 1 - (2 (4 + n))/3, -1], Mod[n, 3] == 2} }], {n, 0, 30}] (* Eric W. Weisstein, Sep 10 2013 *)
Comments