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A094007 Numbers n such that the denominator of the n-th convergent of the continued fraction expansion of e is prime. 5
3, 5, 8, 14, 20, 35, 41, 65, 239, 269 (list; graph; refs; listen; history; text; internal format)



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))).


E. B. Burger, Diophantine Olympics ..., Amer. Math. Monthly, 107 (Nov. 2000), 822-829.

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).


Table of n, a(n) for n=1..10.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality

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.

Eric Weisstein's World of Mathematics, 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.


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


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

Sequence in context: A268515 A070948 A141739 * A159914 A153251 A229167

Adjacent sequences:  A094004 A094005 A094006 * A094008 A094009 A094010




Jonathan Sondow, Apr 20 2004; corrected Apr 21 2004


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



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Last modified August 22 21:19 EDT 2017. Contains 290952 sequences.