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

Author

Jonathan Sondow, Apr 20 2004

Keywords

• nonn

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]

Examples

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

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.

Crossrefs

Cf. A094007.

See also A000040, A000720, A007677, A102049.

Programs

Mathematica

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

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 */

Formula

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