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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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8
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3, 7, 17, 41, 239, 577, 665857, 9369319, 63018038201, 489133282872437279, 19175002942688032928599, 123426017006182806728593424683999798008235734137469123231828679
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OFFSET
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1,1
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COMMENTS
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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.
Is this sequence infinite?
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REFERENCES
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Prime Obsession, John Derbyshire, Joseph Henry Press, April 2004, p 16.
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LINKS
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FORMULA
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MATHEMATICA
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Select[Numerator[Convergents[Sqrt[2], 250]], PrimeQ] (* Harvey P. Dale, Oct 19 2011 *)
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PROG
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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) 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
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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