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

%I A086785 #20 Dec 23 2018 12:40:00
%S A086785 3,103993,833719,4272943,411557987,
%T A086785 7809723338470423412693394150101387872685594299
%N A086785 Primes found among the numerators of the continued fraction rational approximations to Pi.
%C A086785 The numbers listed are primes. For m <= 10000 the only occurrence where both numerator and denominator are prime is 833719/265381.
%C A086785 The next term has 123 digits. - _Harvey P. Dale_, Dec 23 2018
%H A086785 Joerg Arndt, <a href="/A086785/b086785.txt">Table of n, a(n) for n = 1..15</a>
%H A086785 Cino Hilliard, <a href="http://groups.msn.com/BC2LCC/continuedfractions.msnw">Continued fractions rational approximation of numeric constants</a>. [needs login]
%e A086785 The first 4 rational approximations to Pi are 3/1, 22/7, 333/106, 355/113, 103993/33102 where 3 and 103993 are primes.
%t A086785 Select[Numerator[Convergents[Pi,100]],PrimeQ] (* _Harvey P. Dale_, Dec 23 2018 *)
%o A086785 (PARI) \\ Continued fraction rational approximation of numeric functions
%o A086785 cfrac(m,f) = x=f; for(n=0,m,i=floor(x); x=1/(x-i); print1(i,","))
%o A086785 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,",")); ) }
%o A086785 (PARI)
%o A086785 default(realprecision,10^5);
%o A086785 cf=contfrac(Pi);
%o A086785 n=0;
%o A086785 { for(k=1, #cf,  \\ generate b-file
%o A086785     pq = contfracpnqn( vector(k,j, cf[j]) );
%o A086785     p = pq[1,1];  q = pq[2,1];
%o A086785     if ( ispseudoprime(p), n+=1; print(n," ",p) );  \\ A086785
%o A086785 \\    if ( ispseudoprime(q), n+=1; print(n," ",q) );  \\ A086788
%o A086785 ); }
%o A086785 /* _Joerg Arndt_, Apr 21 2013 */
%Y A086785 Cf. A002485, A224936.
%K A086785 easy,nonn
%O A086785 1,1
%A A086785 _Cino Hilliard_, Aug 04 2003
%E A086785 Corrected by _Jens Kruse Andersen_, Apr 20 2013
%E A086785 Corrected offset, _Joerg Arndt_, Apr 21 2013