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.

5, 13, 41, 1093, 797161, 21523361, 926510094425921, 1716841910146256242328924544641, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013

Offset: 1

Author

Cino Hilliard, Oct 02 2005

Keywords

• easy
• nonn

Comments

Or, A046717(n) is prime.

Is this sequence infinite?

References

• Prime Obsession, John Derbyshire, Joseph Henry Press, April 2004, p 16.

Crossrefs

Cf. A088553. [From R. J. Mathar, Aug 18 2008]

Programs

Mathematica

Select[NestList[(Numerator[#]+4*Denominator[#])/(Numerator[#]+Denominator[#])&,1/1,200]//Numerator,PrimeQ] (* Harvey P. Dale, Jan 04 2024 *)

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+.) }

Formula

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)).

A046717 INTERSECT A000040. [From R. J. Mathar, Aug 18 2008]

Extensions

Edited by N. J. A. Sloane, Aug 23 2008