A007677 Denominators of convergents to e. (Formerly M2343)

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

0,3

Comments

Same as A113874 without its first two terms. - Jonathan Sondow, Aug 16 2006

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

Example

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, ...

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 *)

Crossrefs

Cf. A007676 (numerators of convergents to e).

Cf. A003417 (continued fraction of e).

Sequence in context: A270373 A117764 A113874 * A042773 A042173 A317811

Adjacent sequences: A007674 A007675 A007676 * A007678 A007679 A007680

Keyword

nonn,easy,nice,frac

Author

N. J. A. Sloane, Robert G. Wilson v

Status

approved