login
A086397
Numerators of the rational convergents to sqrt(2) if both numerators and denominators are primes.
4
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
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
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