A086785 Primes found among the numerators of the continued fraction rational approximations to Pi.
3, 103993, 833719, 4272943, 411557987, 7809723338470423412693394150101387872685594299
Offset: 1
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]
Programs
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Mathematica
Select[Numerator[Convergents[Pi,100]],PrimeQ] (* Harvey P. Dale, Dec 23 2018 *)
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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
Comments