A094007 Numbers k such that the denominator of the k-th convergent of the continued fraction expansion of e is prime.

3, 5, 8, 14, 20, 35, 41, 65, 239, 269

Offset: 1

Author

Jonathan Sondow, Apr 20 2004; corrected Apr 21 2004

Keywords

• nonn
• more
• hard

Comments

a(n) is the position of A094008(n) in A007677 (denominators of convergents to e), so A007677(a(n)) = A094008(n). Also, A102049(n) is the position of A007677(a(n)) in A000040 (the prime numbers), so A000040(A102049(n)) = A007677(a(n)).

a(11) > 50000. - Lucas A. Brown, Apr 21 2021

Examples

			The convergents for e are 2, 3, 8/3, 11/4, 19/7, ... and so the 3rd convergent is the first one with prime denominator: a(1) = 3 and the 5th convergent is the 2nd one with prime denominator: a(2) = 5.
		

Links

• E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
• Jonathan 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).
• Jonathan Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.
• Jonathan 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

Crossrefs

Cf. A000040, A000720, A007677, A094008, A102049.

Programs

Mathematica

L = {}; cf = ContinuedFraction[E, 5000]; Do[ If[ PrimeQ[ Denominator[ FromContinuedFraction[ Take[ cf, n]] ]], AppendTo[L, n]], {n, Length[cf]}]; L (* Robert G. Wilson v, May 14 2004 *)

Extensions

More terms from Robert G. Wilson v, May 14 2004