A086395 Primes found among the numerators of the continued fraction rational approximations to sqrt(2).

3, 7, 17, 41, 239, 577, 665857, 9369319, 63018038201, 489133282872437279, 19175002942688032928599, 123426017006182806728593424683999798008235734137469123231828679

Offset

1,1

Comments

Or, starting with the fraction 1/1, the prime numerators of fractions built according to the rule: add top and bottom to get the new bottom, add top and twice bottom to get the new top. Or, A001333(n) is prime.

The transformation of fractions is 1/1 -> 3/2 -> 7/5 -> 17/12 -> 41/29 -> ... A001333(n)/A000129(n). - R. J. Mathar, Aug 18 2008

Is this sequence infinite?

References

Prime Obsession, John Derbyshire, Joseph Henry Press, April 2004, p 16.

Links

G. C. Greubel, Table of n, a(n) for n = 1..18

Formula

a(n) = A001333(A099088(n)). - R. J. Mathar, Feb 01 2024

Mathematica

Select[Numerator[Convergents[Sqrt[2], 250]], PrimeQ] (* Harvey P. Dale, Oct 19 2011 *)

Prog

(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) primenum(n, k, typ) = \yp = 1 num, 2 denom. print only prime num or denom. { local(a, b, x, tmp, v); a=1; b=1; for(x=1, n, tmp=b; b=a+b; a=k*tmp+a; if(typ==1, v=a, v=b); if(isprime(v), print1(v", "); ) ); print(); print(a/b+.) }

Crossrefs

Cf. A001333, A086383, A257553.

Sequence in context: A058351 A244782 A175094 * A020730 A003440 A244455

Adjacent sequences: A086392 A086393 A086394 * A086396 A086397 A086398

Keyword

nonn

Author

Cino Hilliard, Sep 06 2003, Jul 30 2004, Oct 02 2005

Extensions

Edited by N. J. A. Sloane, Aug 23 2008 at the suggestion of R. J. Mathar

Status

approved