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

%I A086395 #25 Feb 01 2024 05:27:29
%S A086395 3,7,17,41,239,577,665857,9369319,63018038201,489133282872437279,
%T A086395 19175002942688032928599,
%U A086395 123426017006182806728593424683999798008235734137469123231828679
%N A086395 Primes found among the numerators of the continued fraction rational approximations to sqrt(2).
%C A086395 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.
%C A086395 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
%C A086395 Is this sequence infinite?
%D A086395 Prime Obsession, John Derbyshire, Joseph Henry Press, April 2004, p 16.
%H A086395 G. C. Greubel, <a href="/A086395/b086395.txt">Table of n, a(n) for n = 1..18</a>
%F A086395 a(n) = A001333(A099088(n)). - _R. J. Mathar_, Feb 01 2024
%t A086395 Select[Numerator[Convergents[Sqrt[2],250]],PrimeQ] (* _Harvey P. Dale_, Oct 19 2011 *)
%o A086395 (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,",")); ) }
%o A086395 (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+.) }
%Y A086395 Cf. A001333, A086383, A257553.
%K A086395 nonn
%O A086395 1,1
%A A086395 _Cino Hilliard_, Sep 06 2003, Jul 30 2004, Oct 02 2005
%E A086395 Edited by _N. J. A. Sloane_, Aug 23 2008 at the suggestion of _R. J. Mathar_