This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A111009 #10 Jan 04 2024 18:24:47
%S A111009 5,13,41,1093,797161,21523361,926510094425921,
%T A111009 1716841910146256242328924544641,3754733257489862401973357979128773,
%U A111009 6957596529882152968992225251835887181478451547013
%N A111009 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 4 times bottom to get the new top.
%C A111009 Or, A046717(n) is prime.
%C A111009 Is this sequence infinite?
%D A111009 Prime Obsession, John Derbyshire, Joseph Henry Press, April 2004, p 16.
%F A111009 Given c(0)=1, b(0)=1 then for i=1, 2, .. c(i)/b(i) = (c(i-1)+4*b(i-1)) /(c(i-1) + b(i-1)).
%F A111009 A046717 INTERSECT A000040. [From _R. J. Mathar_, Aug 18 2008]
%t A111009 Select[NestList[(Numerator[#]+4*Denominator[#])/(Numerator[#]+Denominator[#])&,1/1,200]//Numerator,PrimeQ] (* _Harvey P. Dale_, Jan 04 2024 *)
%o A111009 (PARI) primenum(n,k,typ) = \ k=mult,typ=1 num,2 denom. ouyput 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 A111009 Cf. A088553. [From _R. J. Mathar_, Aug 18 2008]
%K A111009 easy,nonn
%O A111009 1,1
%A A111009 _Cino Hilliard_, Oct 02 2005
%E A111009 Edited by _N. J. A. Sloane_, Aug 23 2008