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Numbers k such that the denominator of the k-th convergent of the continued fraction expansion of e is prime.
5

%I #32 Jul 12 2021 03:35:15

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

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

%C 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))).

%C a(11) > 50000. - _Lucas A. Brown_, Apr 21 2021

%H E. B. Burger, <a href="https://www.jstor.org/stable/2695737">Diophantine Olympics and World Champions: Polynomials and Primes Down Under</a>, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.

%H J. Sondow, <a href="https://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641 (article), 114 (2007) 659 (addendum).

%H J. Sondow, <a href="https://arxiv.org/abs/0704.1282"> A geometric proof that e is irrational and a new measure of its irrationality</a>, arXiv:0704.1282 [math.HO], 2007-2010.

%H J. Sondow and K. Schalm, <a href="https://arxiv.org/abs/0709.0671">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</a>, 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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/e.html">e</a>

%e 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.

%t 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 *)

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

%K nonn,more

%O 1,1

%A _Jonathan Sondow_, Apr 20 2004; corrected Apr 21 2004

%E More terms from _Robert G. Wilson v_, May 14 2004