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

%I A094008 #24 Feb 16 2025 08:32:53
%S A094008 3,7,71,18089,10391023,781379079653017,2111421691000680031,
%T A094008 1430286763442005122380663256416207
%N A094008 Primes which are the denominators of convergents of the continued fraction expansion of e.
%C A094008 The position of a(n) in A000040 (the prime numbers) is A102049(n) = A000720(a(n)). - _Jonathan Sondow_, Dec 27 2004
%C A094008 The next term has 166 digits. [_Harvey P. Dale_, Aug 23 2011]
%H A094008 Joerg Arndt, <a href="/A094008/b094008.txt">Table of n, a(n) for n = 1..10</a>
%H A094008 E. B. Burger, <a href="https://www.jstor.org/stable/2695737">Diophantine Olympics and World Champions: Polynomials and Primes Down Under</a>, Amer. Math. Monthly, 107 (Nov. 2000), 822-829.
%H A094008 J. Sondow, <a href="https://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641 (article), 114 (2007) 659 (addendum).
%H A094008 J. Sondow, <a href="http://arxiv.org/abs/0704.1282"> A geometric proof that e is irrational and a new measure of its irrationality</a>, arXiv:0704.1282 [math.HO], 2007-2010.
%H A094008 J. Sondow and K. Schalm, <a href="http://arxiv.org/abs/0709.0671">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</a>, 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.
%H A094008 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e.html">e</a>.
%F A094008 a(n) = A007677(A094007(n)) = A000040(A102049(n)).
%e A094008 a(1) = 3 because 3 is the first prime denominator of a convergent, 8/3, of the simple continued fraction for e
%t A094008 Block[{$MaxExtraPrecision=1000},Select[Denominator[Convergents[E,500]], PrimeQ]] (* _Harvey P. Dale_, Aug 23 2011 *)
%o A094008 (PARI)
%o A094008 default(realprecision,10^5);
%o A094008 cf=contfrac(exp(1));
%o A094008 n=0;
%o A094008 { for(k=1, #cf,  \\ generate b-file
%o A094008     pq = contfracpnqn( vector(k,j, cf[j]) );
%o A094008     p = pq[1,1];  q = pq[2,1];
%o A094008 \\    if ( ispseudoprime(p), n+=1; print(n," ",p) );  \\ A086791
%o A094008     if ( ispseudoprime(q), n+=1; print(n," ",q) );  \\ A094008
%o A094008 ); }
%o A094008 /* _Joerg Arndt_, Apr 21 2013 */
%Y A094008 Cf. A094007.
%Y A094008 See also A000040, A000720, A007677, A102049.
%K A094008 nonn
%O A094008 1,1
%A A094008 _Jonathan Sondow_, Apr 20 2004