A086791 -id:A086791 - 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)).

A086788 Primes found among the denominators of the continued fraction rational approximations to Pi.

Original entry on oeis.org

7, 113, 265381, 842468587426513207

Offset: 1

Cino Hilliard, Aug 04 2003; corrected Jul 30 2004

The next term is too large to include.

			The first 5 rational approximations to Pi are 3/1, 22/7, 333/106, 355/113, 103993/33102; of these, the prime denominators are 7 and 113.

Joerg Arndt, Table of n, a(n) for n = 1..10
Cino Hilliard, Continued fractions rational approximation of numeric constants. [needs login]

Cf. A086791, A086785.

cfracdenomprime(m,f) = { default(realprecision,3000); cf = vector(m+10); x=f; for(n=0,m, i=floor(x); x=1/(x-i); cf[n+1] = i; ); for(m1=0,m, r=cf[m1+1]; forstep(n=m1,1,-1, r = 1/r; r+=cf[n]; ); numer=numerator(r); denom=denominator(r); if(ispseudoprime(denom),print1(denom,",")); ) }

default(realprecision,10^5);
cf=contfrac(Pi);
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) );  \\ A086785
    if ( ispseudoprime(q), n+=1; print(n," ",q) );  \\ A086788
); }
/* Joerg Arndt, Apr 21 2013 */

Offset corrected by Joerg Arndt, Apr 21 2013

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

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A086788 Primes found among the denominators of the continued fraction rational approximations to Pi.

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