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A086395
Primes found among the numerators of the continued fraction rational approximations to sqrt(2).
8
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
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
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