A086788 - OEIS

%I #33 Nov 23 2019 04:45:10

%S 7,113,265381,842468587426513207

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

%C The next term is too large to include.

%H Joerg Arndt, <a href="/A086788/b086788.txt">Table of n, a(n) for n = 1..10</a>

%H Cino Hilliard, <a href="http://groups.msn.com/BC2LCC/continuedfractions.msnw">Continued fractions rational approximation of numeric constants</a>. [needs login]

%e 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.

%o (PARI)

%o 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,",")); ) }

%o (PARI)

%o default(realprecision,10^5);

%o cf=contfrac(Pi);

%o n=0;

%o { for(k=1, #cf,  \\ generate b-file

%o     pq = contfracpnqn( vector(k,j, cf[j]) );

%o     p = pq[1,1];  q = pq[2,1];

%o \\    if ( ispseudoprime(p), n+=1; print(n," ",p) );  \\ A086785

%o     if ( ispseudoprime(q), n+=1; print(n," ",q) );  \\ A086788

%o ); }

%o /* _Joerg Arndt_, Apr 21 2013 */

%Y Cf. A086791, A086785.

%K easy,nonn

%O 1,1

%A _Cino Hilliard_, Aug 04 2003; corrected Jul 30 2004

%E Offset corrected by _Joerg Arndt_, Apr 21 2013