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

7, 113, 265381, 842468587426513207

Offset: 1

Author

Cino Hilliard, Aug 04 2003; corrected Jul 30 2004

Keywords

• easy
• nonn

Comments

The next term is too large to include.

Examples

			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.
		

Links

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

Crossrefs

Cf. A086791, A086785.

Programs

PARI

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,",")); ) }

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

Offset corrected by Joerg Arndt, Apr 21 2013