A102049 -id:A102049 - cp's OEIS frontend

A094008 Primes which are the denominators of convergents of the continued fraction expansion of e.

3, 7, 71, 18089, 10391023, 781379079653017, 2111421691000680031, 1430286763442005122380663256416207

Offset: 1

Jonathan Sondow, Apr 20 2004

nonn

The position of a(n) in A000040 (the prime numbers) is A102049(n) = A000720(a(n)). - Jonathan Sondow, Dec 27 2004

The next term has 166 digits. [Harvey P. Dale, Aug 23 2011]

			a(1) = 3 because 3 is the first prime denominator of a convergent, 8/3, of the simple continued fraction for e

Joerg Arndt, Table of n, a(n) for n = 1..10
E. B. Burger, Diophantine Olympics and World Champions: Polynomials and Primes Down Under, 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).
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.

Cf. A094007.

See also A000040, A000720, A007677, A102049.

Block[{$MaxExtraPrecision=1000},Select[Denominator[Convergents[E,500]], PrimeQ]] (* Harvey P. Dale, Aug 23 2011 *)

default(realprecision,10^5);
cf=contfrac(exp(1));
n=0;
{ for(k=1, #cf,  \\ generate b-file
    pq = contfracpnqn( vector(k,j, cf[j]) );
    p = pq[1,1];  q = pq[2,1];
\\    if ( ispseudoprime(p), n+=1; print(n," ",p) );  \\ A086791
    if ( ispseudoprime(q), n+=1; print(n," ",q) );  \\ A094008
); }
/* Joerg Arndt, Apr 21 2013 */

a(n) = A007677(A094007(n)) = A000040(A102049(n)).

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

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Apr 20 2004; corrected Apr 21 2004

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

			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.

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

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

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 *)

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

cp's OEIS Frontend

A094008 Primes which are the denominators of convergents of the continued fraction expansion of e.

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