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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. 4
3, 5, 8, 14, 20, 35, 41, 65, 239, 269 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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))).

REFERENCES

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).

LINKS

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.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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

Sequence in context: A095223 A070948 A141739 * A159914 A153251 A229167

Adjacent sequences:  A094004 A094005 A094006 * A094008 A094009 A094010

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Apr 20 2004; corrected Apr 21, 2004

EXTENSIONS

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

STATUS

approved

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Last modified September 16 23:31 EDT 2014. Contains 246831 sequences.