A086397 - OEIS

%I #28 Apr 09 2023 02:16:59

%S 3,7,41,63018038201,19175002942688032928599

%N Numerators of the rational convergents to sqrt(2) if both numerators and denominators are primes.

%C Next term, if it exists, is bigger than 489 digits (the 1279th convergent to sqrt(2)). - _Joshua Zucker_, May 08 2006

%C Are the terms >= 7 the primes in A183064? Is this a subsequence of A088165? - _R. J. Mathar_, Aug 16 2019

%C Yes, the terms >= 7 are the primes in A183064 and are a subsequence of A088165.  a(1)=3 is from the numerator of 3/2, but no other convergents > sqrt(2) can appear in this sequence because they all have even denominator.  All terms >= 7 are lower principal convergents from A002315/A088165/A183064 - _Martin Fuller_, Apr 08 2023

%H Andrej Dujella, Mirela Jukić Bokun, and Ivan Soldo, <a href="https://arxiv.org/abs/1706.01959">A Pellian equation with primes and applications to D(-1)-quadruples</a>, arXiv:1706.01959 [math.NT], 2017.

%t For[n = 2, n < 1500, n++, a := Join[{1}, Table[2, {i, 2, n}]]; If[PrimeQ[Denominator[FromContinuedFraction[a]]], If[PrimeQ[Numerator[FromContinuedFraction[a]]], Print[Numerator[FromContinuedFraction[a]]]]]] (* _Stefan Steinerberger_, May 09 2006 *)

%o (PARI) cfracnumdenomprime(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)&&ispseudoprime(denom), print1(numer",");numer2=numer;denom2=denom); ) default(realprecision,28); }

%Y Denominators are A118612.

%K frac,more,nonn

%O 1,1

%A _Cino Hilliard_, Sep 06 2003

%E More terms from _Cino Hilliard_, Jan 15 2005

%E Edited by _N. J. A. Sloane_, Aug 06 2009 at the suggestion of _R. J. Mathar_