cp's OEIS Frontend

A086397 Numerators of the rational convergents to sqrt(2) if both numerators and denominators are primes.

%I A086397 #28 Apr 09 2023 02:16:59
%S A086397 3,7,41,63018038201,19175002942688032928599
%N A086397 Numerators of the rational convergents to sqrt(2) if both numerators and denominators are primes.
%C A086397 Next term, if it exists, is bigger than 489 digits (the 1279th convergent to sqrt(2)). - _Joshua Zucker_, May 08 2006
%C A086397 Are the terms >= 7 the primes in A183064? Is this a subsequence of A088165? - _R. J. Mathar_, Aug 16 2019
%C A086397 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 A086397 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 A086397 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 A086397 (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 A086397 Denominators are A118612.
%K A086397 frac,more,nonn
%O A086397 1,1
%A A086397 _Cino Hilliard_, Sep 06 2003
%E A086397 More terms from _Cino Hilliard_, Jan 15 2005
%E A086397 Edited by _N. J. A. Sloane_, Aug 06 2009 at the suggestion of _R. J. Mathar_