A110908 Start with the fraction 1/1, list n when the numerator and denominator are both prime for fractions built according to the rule: Add old top and old bottom to get the new bottom, add old top and 6 times the old bottom to get the new top.

1, 4, 52, 106

Offset: 1

Author

Cino Hilliard, Oct 02 2005, Jul 05 2007

Keywords

• more
• nonn

Comments

k is the multiple 6 in the PARI code. The sequence of fractions found with the property that both numerator and denominator are prime is as follows.

n, num/denom

1, 7/2

4, 241/101

52, 15848109838244286131940714481/6469963748546758449049574741

106, 1732765524527243824670663837908764472971413888795440694899 / 7073985631629662697450635044051857198371361627935450689

From Robert Israel, Aug 12 2016: (Start)

n such that A002532(n+1) and A002533(n+1) are both prime.

Note that A002532(n+1) and A002533(n+1) are always coprime, so the fractions are in lowest terms.

No other terms <= 12000.

Heuristically we would expect A002532(n+1) to be prime with probability ~ constant/n and A002533(n+1) to be prime with probability ~ constant/n, so both prime with probability ~ constant/n^2.

Since Sum_n 1/n^2 converges, we should expect this sequence to be finite.

Since A002532(n+1) is divisible by 2 if n is odd and by 3 if n == 2 (mod 3), all terms after the first == 0 or 4 (mod 6). (End)

Examples

			The first four fractions according to the rule are
n,
1,7/2
2,19/9
3,73/28
4,241/101
n=2,3 did not make the list because 9 and 28 are not prime.
		

References

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

Crossrefs

Cf. A002532, A002533.

Programs

Maple

A:= gfun:-rectoproc({a(n+2)-2*a(n+1)-5*a(n), a(0)=1, a(1)=7},a(n), remember):
B:= gfun:-rectoproc({a(n+2)-2*a(n+1)-5*a(n), a(0)=1, a(1)=2},a(n),remember):
select(n -> isprime(A(n)) and isprime(B(n)), [1,seq(seq(6*k+j,j=[0,4]),k=0..1000)]); # Robert Israel, Aug 12 2016

Mathematica

Position[Rest@ NestList[{Numerator@ #, Denominator@ #} &[(#1 + 6 #2)/(#1 + #2)] & @@ # &, {1, 1}, 2000], k_ /; Times @@ Boole@ Map[PrimeQ, k] == 1] // Flatten (* Michael De Vlieger, May 13 2017 *)

PARI

primenumdenom(n,k) = { local(a,b,x,tmp,v); a=1;b=1; for(x=1,n, tmp=b; b=a+b; a=k*tmp+a; if(tmp=1,v=a,v=b); if(ispseudoprime(a)&ispseudoprime(b),print1(x","); ) ); print(); print(a/b+.) }

Formula

Given A(0)=1,B(0)=1 then for i=1,2,.. A(i)/B(i) = (A(i-1)+6*B(i-1))/(A(i-1)+B(i-1)).

A(n) = A002532(n+1). B(n) = A002533(n+1). - Robert Israel, Aug 12 2016