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

3, 103993, 833719, 4272943, 411557987, 7809723338470423412693394150101387872685594299

Offset

1,1

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

Links

Joerg Arndt, Table of n, a(n) for n = 1..15

Cino Hilliard, Continued fractions rational approximation of numeric constants. [needs login]

Example

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

Mathematica

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

Prog

(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 */

Crossrefs

Cf. A002485, A224936.

Sequence in context: A164841 A171366 A292691 * A159577 A116536 A224241

Adjacent sequences: A086782 A086783 A086784 * A086786 A086787 A086788

Keyword

easy,nonn

Author

Cino Hilliard, Aug 04 2003

Extensions

Corrected by Jens Kruse Andersen, Apr 20 2013

Corrected offset, Joerg Arndt, Apr 21 2013

Status

approved