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

3, 7, 41, 63018038201, 19175002942688032928599

Offset

1,1

Comments

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

Are the terms >= 7 the primes in A183064? Is this a subsequence of A088165? - R. J. Mathar, Aug 16 2019

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

Links

Table of n, a(n) for n=1..5.

Andrej Dujella, Mirela Jukić Bokun, and Ivan Soldo, A Pellian equation with primes and applications to D(-1)-quadruples, arXiv:1706.01959 [math.NT], 2017.

Mathematica

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 *)

Prog

(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); }

Crossrefs

Denominators are A118612.

Sequence in context: A181148 A179907 A080581 * A229941 A019018 A018993

Adjacent sequences: A086394 A086395 A086396 * A086398 A086399 A086400

Keyword

frac,more,nonn

Author

Cino Hilliard, Sep 06 2003

Extensions

More terms from Cino Hilliard, Jan 15 2005

Edited by N. J. A. Sloane, Aug 06 2009 at the suggestion of R. J. Mathar

Status

approved