A086785 Primes found among the numerators of the continued fraction rational approximations to Pi.

3, 103993, 833719, 4272943, 411557987, 7809723338470423412693394150101387872685594299

Offset: 1

Author

Cino Hilliard, Aug 04 2003

Keywords

• easy
• nonn

Comments

The numbers listed are primes. For m <= 10000 the only occurrence where both numerator and denominator are prime is 833719/265381.

The next term has 123 digits. - Harvey P. Dale, Dec 23 2018

Examples

			The first 4 rational approximations to Pi are 3/1, 22/7, 333/106, 355/113, 103993/33102 where 3 and 103993 are primes.
		

Links

• Joerg Arndt, Table of n, a(n) for n = 1..15
• Cino Hilliard, Continued fractions rational approximation of numeric constants. [needs login]

Crossrefs

Cf. A002485, A224936.

Programs

Mathematica

Select[Numerator[Convergents[Pi,100]],PrimeQ] (* Harvey P. Dale, Dec 23 2018 *)

PARI

\\ Continued fraction rational approximation of numeric functions
cfrac(m,f) = x=f; for(n=0,m,i=floor(x); x=1/(x-i); print1(i,","))
cfracnumprime(m,f) = { cf = vector(100000); 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(isprime(numer),print1(numer,",")); ) }

PARI

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 */

Extensions

Corrected by Jens Kruse Andersen, Apr 20 2013

Corrected offset, Joerg Arndt, Apr 21 2013