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

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

Offset

1,1

Comments

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]

Links

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.

Formula

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

Example

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

Mathematica

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

Prog

(PARI)
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 */

Crossrefs

Cf. A094007.

See also A000040, A000720, A007677, A102049.

Sequence in context: A359748 A113841 A128072 * A209477 A209336 A078552

Adjacent sequences: A094005 A094006 A094007 * A094009 A094010 A094011

Keyword

nonn

Author

Jonathan Sondow, Apr 20 2004

Status

approved