A086788 Primes found among the denominators of the continued fraction rational approximations to Pi.
7, 113, 265381, 842468587426513207
Offset: 1
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]
Programs
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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
Comments