

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.


0




OFFSET

1,2


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)


REFERENCES

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


LINKS

Table of n, a(n) for n=1..4.


FORMULA

Given A(0)=1,B(0)=1 then for i=1,2,.. A(i)/B(i) = (A(i1)+6*B(i1))/(A(i1)+B(i1)).
A(n) = A002532(n+1). B(n) = A002533(n+1).  Robert Israel, Aug 12 2016


EXAMPLE

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.


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 *)


PROG

(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+.) }


CROSSREFS

Cf. A002532, A002533.
Sequence in context: A182044 A000854 A232517 * A232507 A233474 A101354
Adjacent sequences: A110905 A110906 A110907 * A110909 A110910 A110911


KEYWORD

more,nonn


AUTHOR

Cino Hilliard, Oct 02 2005, Jul 05 2007


STATUS

approved



