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A086395 Primes found among the numerators of the continued fraction rational approximations to sqrt(2). 7
3, 7, 17, 41, 239, 577, 665857, 9369319, 63018038201, 489133282872437279, 19175002942688032928599, 123426017006182806728593424683999798008235734137469123231828679 (list; graph; refs; listen; history; text; internal format)
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 -> ... where the numerators are A001333. - 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

Given c(0)=1, b(0)=1 then for i=1, 2, .. c(i)/b(i) = (c(i-1)+2*b(i-1)) /(c(i-1) + b(i-1)).

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.

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

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Last modified November 26 03:20 EST 2020. Contains 338632 sequences. (Running on oeis4.)