A086791 Primes found among the numerators of the continued fraction rational approximations to e.
2, 3, 11, 19, 193, 49171, 1084483, 563501581931, 332993721039856822081, 3883282200001578119609988529770479452142437123001916048102414513139044082579
Offset: 1
Examples
The first 8 rational approximations to e are 2/1, 3/1, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71. The numerators 2, 3, 11, 19, 193 are primes.
Links
- Joerg Arndt, Table of n, a(n) for n = 1..11
- Cino Hilliard, Continued fractions rational approximation of numeric constants.
Programs
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PARI
\\ Continued fraction rational approximation of numeric constants f. m=steps. cfracnumprime(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(numer),print1(numer,",")); ) } -
PARI
default(realprecision,10^5); cf=contfrac(exp(1)); 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) ); \\ A086791 \\ if ( ispseudoprime(q), n+=1; print(n," ",q) ); \\ A094008 ); } /* Joerg Arndt, Apr 21 2013 */