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A086395
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Primes found among the numerators of the continued fraction rational approximations to sqrt(2).
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3
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3, 7, 17, 41, 239, 577, 665857, 9369319, 63018038201, 489133282872437279, 19175002942688032928599, 123426017006182806728593424683999798008235734137469123231828679
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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 -> 41/19 -> ... where the numerators are A001333. - R. J. Mathar, Aug 18 2008
Is this sequence infinite?
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REFERENCES
| Prime Obsession, John Derbyshire, Joseph Henry Press, April 2004, p 16.
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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)).
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MATHEMATICA
| Select[Numerator[Convergents[Sqrt[2], 250]], PrimeQ] (* From Harvey P. Dale, Oct 19 2011 *)
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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+.) }
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CROSSREFS
| Cf. A086383.
Sequence in context: A131721 A058351 A175094 * A020730 A003440 A102071
Adjacent sequences: A086392 A086393 A086394 * A086396 A086397 A086398
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KEYWORD
| nonn
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AUTHOR
| Cino Hilliard (hillcino368(AT)gmail.com), Sep 06 2003, Jul 30 2004, Oct 02 2005
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Aug 23 2008 at the suggestion of R. J. Mathar
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